Most of these things are intended for those renting their own places, but many will also apply to those taking school-furnished housing...Note that if you have a school-provided apartment, you won't have to deal with many of the items below. On the other hand, you will be dependent upon a Chinese school owner for your housing.

You'll also have to live with their choice of apartments.

I'd wager most of us begin our time in China with a school apartment, but if you stay around a while chances are good you will eventually want your own place.
Foreigners are required to live in housing approved by the local police, and you're required to register your address with the police within 30 days of moving into a place. It's not uncommon for a PSB police mans to come inspect your apartment to make sure you'll be safe there. This usually means that you'll need iron grates over the windows and a good secure door. Your employer's foreign affairs person should be able to help you navigate these dealings.
Chinese leases are usually at least 3 months, with 6-month seen sometimes and 12-month the most common. Month-by-month is possible but can be very hard to find! Typically, Chinese landlords will want to be paid in full for the entire lease- as in up to 12 months' rent- plus a deposit. This is often negotiable- some landlords will let you pay monthly as long as you sign a longer lease. In other cases, your school may pay the lease for you, then keep your housing allowance and/or deduct the amount from your pay. Your lease is probably going to be in Chinese- ask someone to go through it with you before you sign.
Move-in can be very pricey in China. Many landlords will want 3-12 months of rent, 1-2 months of rent as a deposit...plus the broker fee if you used a broker. (Broker fees range from 35-100% of rent in various cities) Sometimes schools will help you with this (at least as a loan) but not all will. Many landlords will come down to 1-2 months rent and 1 month deposit...again plus broker fee if applicable. There are many variables but be prepared to lay down US $800-1200 move-in cost for a 2000-RMB-per-month apartment if you finance it all yourself.
I can't possibly cover the range of possibilities on apartments here. Even if I had the time I'm not sure I have the words. It won't be like a house, or maybe a Western apartment, but it should be big enough for you to live in, with some adjustment on your part...and it'll often be a palace compared to what the average Chinese family lives in. ON AVERAGE, but not always, a Chinese apartment will run noticeably smaller than a Western apartment. Even in a spacious apartment here, be prepared for the kitchen and bathroom to appear impossibly small to you. You get used to it.
In China there are no single-family houses, so locals typically buy an apartment here. The apartment you rent is likely an investment or someone's eventual retirement home. As privately-owned property, they can be different in size and definitely different in appearance and appointments. The inside of every apartment is totally different from any other.
In older buildings, the building outside your apartment may be a frightful dump- dirty, unpainted, unfinished, exposed pipes and wires, poor or no hallway/stairway lighting, piles of trash in the lobbies, etc. Newer buildings will be nicer and more like a Western apartment building. In both cases the apartments can be a far cry from the building in appearance, running from toilets to mansions. At their best they resemble upscale Western apartments. At their worst they are cramped, grimy, dingy Stalinist warrens. Personally, I've only had one really nasty apartment in all my time here...all the others were really rather pleasant once I got used to the way things are done here.
Older apartment buildings run 5-7 stories and often do NOT have an elevator. Newer buildings tend to be high-rise and these always have an elevator. If there is an elevator, it WILL be slow- there's never enough elevator capacity to serve the number of tenants in the building. Even if there are multiple elevators, everyone will always hit all the buttons, both up and down, so every elevator always stops on every goddam floor. Even with this, you probably DO want an elevator. Even if you're healthy enough to haul up and down stairs all day, do you really want to lug every grain of rice you ever buy up 5 flights of stairs? When you travel or move, do you want to have to carry everything up and down 5 flights of stairs even once?
Furnished apartments TYPICALLY include of course the basic furniture plus a small burner set for cooking (there are almost no ovens in private homes here, although small electric ones can now be bought) which you should make sure is fitted with an exhaust hood, a TV, a washing machine (in China a "clothes dryer" is a small rack fitted with pins for hanging things, or a bamboo pole 4-5 meters long upon which clothes are hung), and a telephone. There may or may not be a bottled-water dispenser or a microwave, but these things are inexpensive here and can easily be negotiated. There will be a Western-style toilet. There will also be a shower, but here the term "shower" can mean anything from a glass-walled area like in the West to a showerhead on the bathroom wall and a drain in the middle of the bathroom floor. There will also be a water heater...this can be anything from a flow-through gas heater (best!) to a wall-mounted electric tank like a small version of a Western insulated water heater (OK but they run out fast) to a small electrically-heated holding tank that feeds out the water simply by means of gravity. (Run! Run like the wind and never look back!)
Your apartment should be air-conditioned. These are typically a wall-mounted unit connected to a compressor unit outside your apartment. Most can both cool AND heat, but you might want to make sure before signing a lease. These are basically one-room units...I LOVE air-conditioning, and in my opinion you should have a separate unit for each bedroom plus the living room, or you should negotiate this with the landlord, or you should walk away. It's STINKING hot here in the Summer, especially in the South. In north China, there will be steam radiators for heat as well...in the far north it's common to see heat but no AC. However, radiators alone will not keep you very comfortable if the weather is really cold...and winters in much of China can be SERIOUSLY cold.
It's possible that your apartment will have added amenities- extra TVs in some bedrooms, DVD players, nicer decor, and so on, but this is entirely at the discretion of the landlord.
Things you'll need to buy: Dishes, towels, bedclothes, electric fans. Electric space heaters-even the deep South can have terrible cold snaps, and even with the radiators the North may have apartments colder than you're used to. A DVD player is virtually a must. You may also need more bins/boxes/etc. for storing things.
Things to watch for:Consider avoiding a ground-floor apartment. They're noisier, less private, and more crime-prone than higher floors. IMHO 2nd floor is primo; 3rd floor is tolerable if the apartment is nice. Anything higher without an elevator is too much work...and if I wanted to work, I'd get a job.
Make sure the landlord will cover the part of the first utility bill dating from before you moved in.
Make sure the heat/airconditioning is adequate for the apartment. You will likely want more than one bedroom to be air-conditioned in order to be comfortable! If it's not there, make installing new AC a condition for taking the apartment. If they won't make it adequately comfortable, walk away. There are other places.
Make sure all the windows are screened and that the screens are in good repair- China has enormous, muscular, vicious, foul-tempered mosquitos the size of cocktail shrimp.
Make sure that everything is in good working order before signing a lease.
Also before signing the lease, sit down with your landlord and a witness such as your broker or school foreign liaison, and make it explicitly clear that you will replace things if YOU break them, but if the item simply dies then HE/SHE is responsible for fixing or replacing them. If possible, get this in writing. I once had a TV die of old age after I'd lived in an apartment nearly a year, only to have the landlady TRY to insist that I should go buy her a new one. (She eventually backed down...)
Make sure that a circuit-breaker is installed that's big enough to handle all your air-conditioners at once, especially in older apartments...foreigners tend to use more juice than Chinese. Even with this, you'll still probably have a learning curve of how many things you can turn on at once before all the power goes out.
Foreigners often have more belongings than Chinese- make sure the storage in the apartment is adequate.
Make sure someone shows you how to use the stove, air-conditioners, washing machine, water heater, etc. You ain't in Kansas anymore...it's different here.
Make sure you know how bills work. It's common for someone to stop by once a month to read your meters. Some places send bills in the mail, while others expect you to simply show up each month and pay some money. You can usually pay bills in banks; in Shanghai you can also pay them in some convenience stores. Never take a place where they come knocking on your door to collect utilities...you can't imagine the horror.
Make sure you know where the circuit-breakers are. Many apartments have TWO sets...ask!
Find out about fees for your apartment building, if any. I lived in one place where you had to pay 0.5 RMB to go up in the elevators...tenants could buy a pass but visitors had to pony up. Some buildings have people who will come up once a month, hack their way through the filth in the hallways, and knock on your door to collect a couple of RMB for a "sanitation fee". These fees are usually tiny- literally pennies- but the mindless drones hired to collect them will make your life a living hell until they get it. My policy is often to insist that friends call before coming if they want to come inside...and I would ignore any knock on the door I wasn't expecting. It's not easy- Chinese people can happily knock on your door for 5 hours to collect their 40 cents. Stereo headphones are your friends.
Speaking of headphones, noise is one of the worst problems many people face in apartments here. Chinese people often seem to assume that if it's worth saying at all, it's worth saying at a volume that will part your hair. In some areas firecrackers are lit quite often, especially for move-ins and celebrations. Worst of all, apartment owners are legally entitled to do any work upon their homes they want to; if you're lucky there'll be a legal time limit such as 8am to 10pm. This means people can freely hammer, saw, run power equipment, etc. for 14 hours or more a day, for weeks at a time.. There's nothing you can do but grin and bear it. I'm large and intimidating-looking; sometimes threats of physical violence have brought fast, temporary relief. I've also confiscated hammers and other tools until at least after sunrise.

But be aware that there are legal and personal risks in taking these actions...tempting though they will be.
Be prepared for attempts to rip you off for your deposit when you move out....have an exit strategy. But be fair- or you have no right to ask them to be.
I'm sure the Barflies will have more tips and horror stories for you!