Endurance affects your ability to resist shock, focus abilities and being dazed by physical attacks (not by a feint, though). High levels of endurance make diseases develop a lot slower and lets you have a higher chance of resisting drug and poison effects, though they are still effective if you fail the first resist roll. Endurance also makes your muscles more resistant to strain, so a character with high endurance can push their muscles harder on the gym without having to worry about muscle strain.
High endurance lets you wield heavy melee weapons like spears easier and able to beat people up wih your fists, so spears and fists are a popular choice for endurance based characters. Since endurance is also one of the stats that affect focus, endurance characters tend to have high focus and make good use of focus abilities. This is why endurance is so important for abominations, because it gives them an alternative approach to gameplay, which is a blessing considering how the mutation limits them regarding the all important brain based skills like medic and repair. Contrary to popular belief, endurance does not give you extra soaks (as of now), but it makes you less likely to get a broken bone. Its debatable whether endurance makes you more resistant to endocrine boosters or not.
Common misconceptions: Having more endurance doesn't increase the amount of drugs a character can take. All characters will overdose at the same amount unless one is an abomination, however, endurance does significantly decrease the chance of having a heart attack while on lots of drugs, as well as increase the chance to resist withdrawal symptoms and debuffs. Endurance also has no effect towards resisting power attack stuns.
Where to train
On rowing machines in various gyms.