1) Ask yourself, realistically, how much room you need to comfortably live with the possessions you currently have and/or the possessions you intend to buy immediately after moving in. Try to avoid renting too much or too little floor space. Measure your furniture and chart it out on graph paper, then look online for apartments with published floor plans if need be/if at all possible.
2) Find out what the maintenance policies are for the complex. Is there a maintenance person/team on the premises? Will the apartment complex cover minor repairs from wear and tear, or will you be expected to cover them?
3) When you've found some potential places, look them up on Google to see if there are any reviews from current or former tenants.
4) What
Jay said.
5) After deciding that an apartment meets your potential requirements, inspect everything very carefully - can be done while taking those pics Jay mentioned. Look for any signs of mold, mildew, water damage, etc.
6) Try to see if the apartment managers or rental agents will let you view/visit the apartment during the evening or weekend, when people are generally home, so as to better judge the amount of noise you're likely to get from other apartments and/or the environment (roads, etc.).
7) Find out what the pet policy is, even if you don't have or want pets.
8) Check the tops of the kitchen cabinet doors to see if they're clean, to determine how well the cleaning crew did their jobs. I missed this when moving into my current apartment, and when I got around to cleaning them the first time, they were far stickier than they should have been for the length of time I'd been living there.
9) Ask management if they know what the average utility bills are for residents of the complex, so you aren't surprised by the amount of your bills. High winter utility bills can also be a sign of shoddy insulation and construction. My electric bills are more than double the summer cost during winter, mainly because the windows and insulation around the windows are pure crap, and the fireplace and chimney ain't helpin' matters, even with the flue closed.
10) Always check for applicable transit options/commute times to wherever you work, assuming you work outside your home. How long are you willing to spend getting to and from work each and every day?
11) Try to find an apartment with a decent selection of grocery stores, or at least with more than one, nearby. The same goes for entertainment and dining options.
There's probably more, if I think of anything, I'll update.