Can you simply buy a clothes dryer and plug it into a wall socket (at an apartment), and dry your clothes?

Yes.  google  110 volt clothes dryer.  You'll find some that are NOT vented.  They'll be small, you might have to dry 1 sheet and a time and probably not blankets at all. 

if it's not too many items, you can hang them on a rack (that's what you are probably doing now)--if not dripping, for that can ruin a floor. For a dryer machine you need the correct circuit 220 as others have said, unless you buy one that is 110/120 which I expect might be more expensive.

I hang my laundry outside on the line, and also use an indoor rack, all through the winter. The trick is NOT to wash too things at a time.

smaller appliances can be had cheapo in thrift shops,tho a lot are closed because of the bug and local taxes

 ma used a lil thing on wheels to push around had the vent hose end in an included plastic small bucket filled with a couple cups of water to catch the lint...a hose extension can be bought to direct the hot air out a window

Yes, if the right kind of outlet is there, as well as an exhaust vent. Standard dryers use 240 volts, two entire circuits, whereas a normal outlet uses 120 volts. The exhaust vent is so your apartment doesn't get full of lint and warm moist air. There are things you can use to catch the lint, but that doesn't do anything about the warm air which may give you a lot of condensation inside the house.

Yes you can.  Buy the dryer and have it delivered.  Then ask the landlord how to plug it in.  Easy-peasy.

Yes you can buy a 120V clothes dryer.

NO you do not have to vent it outside. There are lint catching devices you can put on the dryer vent outlet.

Now for the problems:

On a real dryer, you can dry most anything in 30-45 min.

On the 120V one it can take 2-3 HOURS to dry and they don't hold much laundry.

Without venting outside, you will have much higher humidity in the house unless you open a window or vent it outside. Can be a mold issue after a while.

Plus the added heat you need to air condition against.

And then there is the higher fire risk as most people refuse to have their dryers professionally cleaned and inspected at least annually.

Now for the legal aspect - check your lease to see if you are allowed a clothes dryer in your apartment (or even a washing machine for that matter).

If not, the landlord can legally require you to either remove it, pay more rent, or have you evicted through the courts.

Here is what others have done in the past:

They either bought or made an adapter that goes from the electric stove outlet and converts it to a standard dryer outlet, then they use a real dryer.

BUT the above information would still apply.

Venting they either bought a window vent kit or simply stuck a few feet of dryer hose out a window (dead giveaway that you have a dryer).

There are very small 120 volt dryers available but they still require a exhaust out a window and they do a poor job compared to 240 volt units

You can't. They need a special outlet and a vent to the outside.

Most clothes dryers need a 240 volt outlet.  Standard plugs are 120.  You can't just swap those out as they have to do with the wires coming into the outlet and the breaker it is hooked up to.  There is also the exhaust you'd need to deal with.  You probably could run the hose to a window when in use, but that's not very practical and would likely not work great either.  

Yes, you just plug it in. But you need to put it in the laundry or garage preferably with air vents or open windows. I put mine in a bedroom and the heat and moisture from the machine made everything in there damp. So you need a room where the heat and moisture can escape. 

Edit: in my country, Australia, you just plug them straight into the wall socket, we don't have to worry about voltages. I use my dryer all the time.