The galaxy is filled with all sorts of lovely creatures, and within all of those creatures are some very lovely organs! Once you have these organs, you can but em into whoever you'd like! Lets find out what useful things you can harvest from who. The universe is always changing, so hopefully in the future, there will be even more unique organs across all the different species. This currently only includes round start species and monkeys, but will be updated in the future to contain more
How am I supposed to get at those juicy organs?
Through Surgery of course! In order to extract organs, you are going to have to preform an organ manipulation surgery. This can be preformed on both living and dead targets. Once you get to step 6 of the surgery, you can use a hemostat (or a crowbar or fork in a pinch) to remove all of the organs from the area you want. Its worth noting that external organs that aren't limbs (such as tails) require the feature manipulation surgery, a simple surgery that goes scalpel, retractor, and then hemostat to remove external, or the organ you want to apply. . It is relevant what part of the body you target when you begin the surgery. The chest contains alot of the juicer bits. Its where you will find the heart, stomach, lungs, and liver. The head is also a good spot to crack open. It tends to contain the eyes, ears, tonuge, and importantly the brain of the creature. The groin area usually only contains an appendix, which is mostly useless. You'll be hard pressed to find any interesting organs in the limbs, unless of course your target has some sort of bionic implant in there.
Wait, organs go bad?
Its important to note that non-robotic organs decay over time, even inside of a dead body. There are only a few ways to keep organs fresh.
Stuff them in a living body - Using the body of a monkey or other willing participant, you can rip the organs out of them that you want to make space for. For example, you could take out their liver and install a cool new alien one. Be aware that preforming a living heart transplant without killing the subject requires skill, and doing so with a brain is impossible with current technology. This not only has the benefit of keeping the organ fresh, but also will heal the organ of any previous decay it would have suffered over time (provided it hasn't decayed completely). Just make sure that your subject doesn't run off with all the cool new stuff you put in them.
Stored in a dead body - Dead bodies are usually easier to keep track of than living ones. However, a dead body can not keep an organ fresh on its own. Injecting a body with Formaldehyde, buckling it to a Stasis Bed, or putting it in a Morgue tray will stop the organs inside from decaying. This is a very common tactic.
Cold storage - Ironically has nothing to do with the temperature of the environment. You can keep the organs living and fresh outside the body with an Organ Transport Box, which can be found in medical vendors for a hefty price, or in a smart organ storage fridge common around med bays. As of right now, those are the only ways to keep an organ fresh outside of the body.
Restoring a decayed organ - Organs become useless when they have fully decayed. There is only one way to restore them. By placing them into a body and preforming a corresponding Surgery you can restore them to an operable state once per organ. for lungs, this is a Lobectomy, for hearts, this is a Coronary Bypass, for livers this is a Hepatectomy, for stomachs this is a Gastrectomy. Brains may be healed as many times as needed through Brain Surgery. All of these surgery's carry a chance of failure, and usually have catastrophic results in the process. Certain Chemicals may be able to restore health to certain organs, and Mannitol may be applied directly to a disembodied brain to heal it.
Human
It can be assumed that if a creature does not have a unique organ in a position that it has an organ that is practically the came as that of a human. For example, the livers of humans and the livers of lizard people are practically indistinguable. As the list continues, if I don't list another organ in place of one of these organs assume that it is the same as the baseline human's.
Head 
Brain
The human brain is the baseline. Its where your Soul resides. Wherever your brain is, you will follow. The human brain gives the ability to read and write, as well as the ability to use tools in a complex way. This includes constructing, repairing, and other important stuff for operating on a space station. Placing a human brain into a Less developed body will allow it to do all the things a human can do. A body can not live without a brain, and every creature will have one in some form or another.
Eyes
They let you see, without em you can't. Damage can cause you to become short sighted, and intense damage can cause you to go blind. Bright lights can damage your eyes. Every creature will have eyes in some form or another.
Ears
They let you hear, without em you can't. Damage can cause you to become temporarily blind to to ear ringing, and intense damage can cause you to go completely deaf. Loud noises can damage your eyes Every creature tends to have ears in some form or another. (sense a pattern?)
Tongue
Pretty simple little organ. If you don't have it, you won't taste anything, and you'll have trouble talking. A tongue determines what language you have the physical capability to speak, but not nessicarly the languages you know. A lot of languages require a specific tonuge to not speak in an accent.
Torso 
Heart
Your heart pumps blood. If you don't have it, you'll die pretty quick. Heart damage doesn't effect much until the heart fails completely, which will result is pretty fast death. The heart is also one of the quickest organs to decay, so handle with care. Creatures who do not have blood will not have hearts.
Lungs
Needed to breath! Without em, you will take oxygen damage as if there was none available. Damage to lungs doesn't effect much other than occasional coughing. Human lungs need 16 kPa of oxygen in the environment to function, and can't withstand more than 10 kpa of co2. Human lungs are susceptible to the effects of Plasma, BZ, N2O, Zauker, Freon, Miasma, Pluoxium, and Healium in very low doses. Damage increases how much oxygen you take in. Creatures who don't breath won't have lungs.
Stomach
Needed for digestion, pretty cool. Significant stomach damage will cause you to vomit based with increasing frequency based on the amount of nutriment in your system. The more damage you have the harder it is to keep it down. You won't be able to keep any thing down at all if your stomach is failing. Some creatures will still have stomachs even if they don't need to eat, but they usually have some other purpose instead.
Liver
The livers main function is to filter Toxins out of your system. A damaged liver will remove less toxin over time. A liver that is either not present or non functional will deal toxin damage, give you the inability to process chemicals, and deal damage to your other organs. Creatures who do not metabolize chemicals will not have livers.
Groin
Appendix
This is a useless piece of trash organ. Its doesn't even have a use as a snack for a lizard because its just filled to the brim with bile. Its actually a danger to its owner, as appendicitis can really ruin your day. Appendices can (very rarely) randomly fail. A failed appendix causes toxin damage, stunning, and vomiting. If your goal is to make an organic super soldier, leave this organ out of the mix. Creatures who don't eat won't have Appendices.
Lizard
Lizard men don't have a ton of unique organs, and are otherwise similar to baseline humans.
Forked Tongue
Replacing the normal tongue is a cool reptilian one. This one can speak draconic (language of the lizards) without an accent, but has a hissing accent when speaking galactic common. It also has a slightly higher taste sensitivity.
Lizard Tail (New Groin Organ!)
The tail, besides its cool looks, gives it user a +1 defense against tackles while it is wagging. Lizards however, also start with an innate trait that gives them a -1 defense against tackles while they don't have a tail.
Cosmetic organs
May or may not exist based on the lizard's character creation. This includes the lizard's horn, frill, and snout on their head, as well as their ridge in there chest. These have no effects, but might be just the style you are going for if you want to implant them in yourself.
Mothman
Mothmen only have a few organs of note, and fewer still of benefit.
Moth Eyes
Large insectoid eyes with bulging pupils, disgusting. This will give you a -1 penalty on your bright light protection. This means that sunglasses will not be enough to protect you from flashes, and if you have another penalty to this value (for example night vision goggles) you will be flashable from all directions. There is no positive to having them, but you might be able to implant them into a prisoner to better keep them in line.
Moth Wings (New Chest Organ)
Wings! You can't just fly whenever with these, but in a situation without gravity but still an amount of air pressure you will be able to sore around the station. The down side of this is being on fire or being extremely hot will cause your wings to be burnt off! These wings can also soften falls from high places.
Cosmetic organs
Mothmen might have antennae in their head. These do nothing but you can take em off if you really would like to.
Felinid
Lets just get this over with.
Cat Ears
A replacement for normal ears, these take twice as much damage from loud noises on an account form being hyper sensitive. In addition to this, you can physically see two cat ears sticking out of the wearers head Inu Yasha Style. This may be a positive for you if you are incredibly lame.
Cat Tail (New Groin Organ!)
Its a cat tail that sticks out of your ass, the same color as your hair. You can wag it, but unlike the lizard's tail this has no effect other than perhaps displaying your excitement. If the original owner loses it, they'll be pissed, so good luck finding a spare one for your mad experiments.
Ethereal
Ethereals have unique, electric, biology. Because of this they have a lot of unique and useful organs. (With the notable exception of a perfectly normal liver)
Aeration Reticulum
These are the ethereal's lungs. They have a higher threshold for when the air they breath starts to become too hot, in line with their natural resistance to a hot temperatures. They also hold a unique respiratory pattern, where they can hydro-electrolyze breathed in water vapor into Oxygen and Hydrogen. This is the only way to make hydrogen without machinery from Atmospherics. They are a very good set of lungs.
Biological Battery
This is the ethereal's "stomach" of sorts. It removes the traditional hunger in a creature and replaces it with a thirst for electricity. This is what gives ethereals the ability to drain and give power from APCS and other structures, as well as absorbing some on electric shock. For more information, see Charge mechanics
Crystal Core
This is the ethereal's heart, and what gives ethreals the ability to revive after death once. After being left alone for a while dead, crystals will begin to form on you these can be broken gently if you are disarmed delaying your resurrections, or broken entirely if you take 30 or more damage in this state. Eventually, you fully crystalize immersing yourself into crystal cocoon. Soon enough you will emerge alive and healthy, with the exception of a permanent a new permeant brain trauma.
Electrical Discharger
The ethereal's tongue. It has a more sensitive taste due to a complex vaporization based olfactory system. It is the only tongue able to pronounce voltaic, which is ethereal language.
Plasmaman
Skeleton men that burst into flames have a pretty complicated biology. Go figure. Plasma men have no hunger and therefor lack an appendix.
Plasma Filter
This is the Plasma man's respiratory system. It is comparable to a human pair of lungs, with the very notable exception that 4 kPa plasma is required to breath, and any amount of oxygen will suffocate you. And naturally, you take no damage from inhaling plasma.
Reagent Processing Crystal
This is their liver! You would think it would allow them to metabolize liquid plasma without any harm or something, but in actuality, it does nothing. Its likely to have some sort of effect in the future.
Digestive Crystal
A stomach. This digests things more efficiently than a normal stomach, which chemicals get from your stomach into your bloodstream faster. You can also heal brute damage and cure broken bones by drinking milk.
Plasma Bone Tongue
This Tongue can speak calcic, which is the skeleton language. It is otherwise uninteresting.
Monkey
Monkeys are nearly identical to humans organ wise.
Primate Brain
These are just like human brains, except they don't give there owner the ability to read or use advanced tools. They also control the monkey's tendency to trip over others. A monkey brain is a bad place to be.
Monkey Tail
Just a normal tail, nothing too interesting about it. Monkeys will be upset if they loose it though. Not in a "bite your face off" kind of upset, just a bit sad.
More to be added
I've only covered the round start species so far, but there are many more exotic specimens out there to harvest. Keep up the Frankenstein-ing and the other mad experiments. - Easy