Fitted is the word you are looking for, a fitted sheet is the sheet above the mattress that has extra fabric to extend around the mattress and an elasticized outer edges to fit around the mattress holding it in place.
The other sheet is a flat sheet referred to as a standard bed sheet or drop sheet or top sheet.
When you are putting fresh linens on a bed, you are making up a bed.
The opposite is to strip (down) a bed.
The total effect of putting a bed together (sheets, pillows, covers) is called dressing a bed
When you are tidying up a bed after sleeping in it, you are making a bed.
The common command by parents
Go make your bed!
Before going to sleep, some hotels will turn down your bed by folding back the covers and (if you're lucky) leaving a chocolate on the fluffed pillows.
If you are in a mischievous mood, you can short sheet a bed; a common prank played on newlyweds.
We spend about a third of our life on top of a fitted sheet (usually), this intimate and fated relationship has lead to the phrase
You've made your bed, now lie in it
since there's no escaping it.