Mattress: thing you sleep on, makes it soft.

Mattress pad: thing to protect mattress. It can also help smooth the mattress or make it a little softer. You don't generally sleep directly on it because they're not meant to be washed so often/they take longer to wash and dry because they are thicker.

Fitted sheet: You put this over your mattress pad. It is meant to be washed often. You can also use a flat sheet and tuck it, but as you mentioned it comes undone.

Flat sheet: Goes on top of the fitted sheet. You sleep between the flat sheet and fitted sheet. The flat sheet protects your other covers that aren't easily washable.

Duvet: (Usually) fluffy square/rectangle of warmth. Very difficult to wash (should probably be only washed once a year unless your cat or kid pukes/pisses on it).

Duvet cover: Protects the duvet. Easy to wash.

Quilt/comforter/blanket: May be another layer of warmth, again not so easy to wash (which is why you have the flat sheet).

The sheets you come in contact with (flat, fitted) and pillow case should be washed weekly (or bi-weekly... or whatever). The other stuff can go months or longer without washing because you never sleep directly on it.

Here's an example of a sheet set: http://www.amazon.com/Thread-Count-Queen-Egyptian-Pocket/dp/B0039JWJZI/ref=sr_1_1?s=bedbath&ie=UTF8&qid=1367081353&sr=1-1&keywords=fitted+sheet

I don't know what size bed you have or what your budget is or what colour you want... If it fits a deeper mattress it's usually OK (you can also buy clips if you find the sheet slips, which might be easier than finding a fitted sheet that fits your bed exactly). Obviously better materials cost more, but they're also nicer to sleep on.

Source: I sold sheets for a while.