If the applicant admits to an offense, or the recruiter has reason to believe the applicant is concealing an offense, or a record is indicated during the Entrance National Agency Check (ENAC), then the recruiter will request a complete criminal record from local law enforcement agencies.
Some offenses can be waived, and others cannot. Recruiters themselves, do not have waiver approval/disapproval authority. Some waivers can be approved/disapproved by the Recruiting Battalion Commander, other waivers must be approved/disapproved by the Commanding General of the Army Recruiting Command.
It's important to note that applicants who require a waiver ARE NOT qualified for enlistment, unless/until a waiver is approved. The burden is on the applicant to prove to waiver authorities that they have overcome their disqualifications for enlistment, and that their acceptance would be in the best interests of the Army. Waiver authorities will consider the "whole person" concept when considering waiver applications. If a waiver is disapproved, there is no appeal (the waiver process itself is the appeal -- the individual is not qualified for enlistment and submits a waiver request, appealing to Army recruiting authorities to make an exception in his/her particular case).