The largest portion of your day at MEPS is taken up by the medical examination. You'll start by completing a detailed medical history. Your blood and urine will be taken and examined for this and that (including drugs and blood-alcohol level). Your eyes and hearing will be checked. You'll have to do some stupid-sounding things, such as walking while squatting (this is commonly called the "duck-walk.")
Medical Standards for enlistment are set by the Department of Defense, not the Army. The doctors at MEPS will medically disqualify you, if you fail to meet any of the standards. There are two types of disqualification: temporary and permanent. A temporary disqualification means you can't join right now, but may be able to, at a later time. For example, if you just had an operation the week before. A permanent disqualification means that you failed to meet the published standards, and that won't change with time.
If you're permanently disqualified, the Army can choose to waive the medical disqualification. The commanding officer of the recruiting company will determine whether or not a waiver will be submitted. If the commander approves it, the request goes all the way up, winding its way through the command chain, to the top doctor in the Army Recruiting Command (HQ USAREC Command Surgeon). The CS's office has final approval authority. This process can take several weeks (sometimes months).