Make sure that the kitchen has the basics: a kitchen table and the right number of chairs around it. Check the condition of these items; along with the normal rental responsibilities, you are also responsible for the condition of furnished items during the rental term. Make sure that your lease covers normal wear and tear on the furnished items. However, you will still be held responsible for damaged items. Consequently, make sure that the table is in sturdy condition and free of major scratches or damage. The same goes for the chairs. Make sure the kitchen has a microwave if you're paying for a furnished apartment.
The standard furnishings for your common area include at least a couch, usually a lounge chair of some sort and a lamp. Lighting is an aspect of apartments that aren't usually given much thought until the sunsets on the day you move in, so make sure you're getting appropriate lighting. The room should also include a television (unless it explicitly says in the rental agreement that one won't be furnished, in which case you should make sure the premium you're paying for the furnished apartment is appropriate to what you're getting) and a stand for it. Remember to locate the remote control for the television before the apartment manager leaves.
In the bedroom, make sure that you have the bare essentials for getting through the days in a civilized manner. (If you're a starving college student, you can disregard items as you see fit.) These will include a bed with a mattress and box spring. Make sure the bed comes with a box spring, if not for your personal comfort, then to make sure that the mattress itself is still in good condition at the end of the lease term. You don't want to be charged for "damage" to a mattress. A night stand, lamp (remember lighting), dresser with drawers for clothing and a closet that comes with hangers is also appropriate.
The main essentials for the bathroom include a towel rack, which even most unfurnished rentals have, and a shower curtain. Make sure to check the curtain for mold or mildew before the apartment manager leaves.