Philly left out a few things. Yes, they get a choice of Army, Navy, Air Force or Marine. Yes, some can do riot police (and I've heard rumours of firefighting as well) instead of military service. Some that have skills or jobs of civic importance can be excused (been a few cases of singers/actors getting "hired" to do that kind of job to get out of it, they all got caught at sent back to the Army). Actions of national pride/importance can also excuse them of service, as Philly mentioned with the sports team getting the gold medal. PhD students at KAIST are also exempt from service. Those deemed medically unfit for service are also exempt from service. And finally, any man that lives in the DMZ (there is one village on each side) is exempt from service (must have been born there, and live there year round). There is one other way, serve your 26 months in jail. but generally not the best way. Generally those with religious reasons against military service are allowed another form of federal service or put into non-combat roles in the military, but there have been some that took jail time for refusing to serve.
There are a few other, special, cases but those are exceptions and not set rules.
Another note, anyone who is a natural born Korean citizen (even if they don't claim it, all it takes is one parent having been a Korean citizen) must do their service before Korea will recognize that they have given up their citizenship. This means that if they enter Korea, for a period exceeding 180 consecutive days, between the ages of 18-35 (inclusive) they will be drafted.