Look at everything. Seriously, if there's power and running water to the place, flip on every light switch, turn on every faucet and make sure hot water works alright, flush toilets etc. I look behind refrigerators/washers and make sure there's no bugs. If there's a bug problem in the place, you're likely to find signs of it behind things like the refrigerator/washer & dryer/microwave, etc. Check the ceilings in every room for water damage that might be signs of a leaky roof. Take note of any holes on the wall, or any holes that have been patched up.. any damage that the place might have at all. If you do decide to move in, they should give you a form to fill out listing all of the things that are wrong with the place anyway. Make sure you go over everything and list it all because if you don't, when you move out, they'll assume that damage was from you and they'll take it out of your deposit if you're renting. Look at window frames, open cabinets and closets. Especially the under the sink cabinets to make sure of no leaks or other pest problems. Literally look at everything.
Noise levels. Make sure the visit isn't too short or always at the same time of day (if you go more than once). How close is the street? Railway? Are you under a flight path?
Is there an elevator to your floor? Check the lights. Any bulbs burnt out and not yet replaced?
Any signs of graffiti near by?
Are the windows double-glazed? (Saves on heating bills, and reduces outside noise.) Do they open/close easily?
Turn on the taps and check the water. Does it come out clear straight away? Check the ceiling in the bathroom/shower. You're looking for anything that looks like black dots on the ceiling. (Mould from damp.)
Look for mold where there are water pipes (bathroom, kitchen). Check all the cupboards, if they smell funny, there may be rodents or various small critters. Also take a look for small holes in walls, as they are places where rodents or insects can get into the place. Also just basic stuff. Make sure the water flows properly, the stove and oven turn on, etc.
Ask when it was built, that will say a lot. Lots of good advice by everyone else on here - check for an signs of previous damage - cracked or running paint or tiles. I moved into an apt and found out it didn't have grounded electricity :( , common in much older homes but I was shocked.
I went through this about a year ago. If you are allowed to look at the apartment alone (without an employee) take advantage of it! Look at everything. run water, flush the toilets, look for water spots on floor and ceilings, look at window sills (bugs/air water). Ask when they were built, this will help with knowing how stable they are(not that they are going to fall down but its just nice to know) Also look at the carpet, it will tell you a lot about the standards they hold.
My mom's friend works at an apartment complex and she said that if a place does not take care of their dumpsters they don't take care of the complex or people. It seems true to all that I have looked at so if the dumpster is there look at it. If it is dirty and out of line and old, it may not be the right one for you.
Additionally, take a drive around the place before you talk to anyone. Just get a good look all the way around to see what you are getting yourself into. One thing that my friend thinks is a good idea is the apartments that you have to buzz people into. This allows anyone who wants to walk in, not able to get in unless they are buzzed in personally.
Even though you are going to be living there alone take some friends with you! They think differently than you and will look at different things. Also, if someone from the complex goes with you to look at the apartment they will try to distract you from things. Try not to get distracted, I know sometimes it's hard but you honestly have to be looking out for yourself everywhere.
Another thing that I was very interested in was the security. Some places have guards, some have nothing but " neighborhood watch." One place actually had some police officers living there and another had off duty police to come sit at night in the parking lot.
Another thing to think about is parking. Some places could care less how many cars per apartment and some others actually had assigned places.
Good Luck! And I hope you find a place that fits you grandly! :)
The first thing I ask is what utilities are included in rent. Note that if you have a lot of neighbors and most of the utilities are included in the rent your rent will spike if they (or you) happen to be wasteful. On the other hand, you will have the peace of mind knowing that the bills are all combined and all you will have to worry about bill wise is rent. You never know, your neighbors might not be that wasteful. It might be worth the risk.
*Check the condition of the apartment. If there is anything wrong with it, take pictures and bring it up with your landlord before you move in so you are not liable for the damage if and when you move out. Ceilings, floors, carpets, water, etc. Check it all.
*Look at the neighborhood, do you feel safe there? You want to be in a place that's safe for obvious reasons. Always go with your gut about a place.
*Try to look up reviews on yelp! And such. They provide valuable insight on what others think of your prospective place. It's good to take in mind what other tenets say because you never know if that could happen to you or not. Good luck!