I'm a recruiting NCO so let me jump in here!
The US Armed Forces have been all volunteer since the suspension of conscription in the Mid 70's (though males are still required to register when they turn 18).
Each branch of service (Army, Navy, Marines, Coast Guard, Air Force) sets its own enlistment standards (with the exception of medical standards which are shared) age, aptitude, moral, etc.
Currently the Army is not considering any applicants with any felony charges on their record. We can enlist applicants with 1 or 2 misdemeanors (more than 1 requires a waiver), though certain crimes are non-waiverable. Certain positions require either secret or top secret security clearances, and for those positions you need a pretty spotless record (to include credit history).
Things that will prevent you from enlisting: any felony, any domestic violence charge, any sex crime, and drug charge other than simple possession of marijuana, more than 3 misdemeanors of any kind, more than 1 charge of driving while intoxicated, etc.
The days of a Judge telling someone "go to war or go to jail" are over.
The Army did a study a few years ago which concluded of the "target demographic" (males 17-25) only ~30% of Americans were eligible to enlist based on medical, education, moral and aptitude standards.
SSG Gunn