I went to IKEA and got me a new sheet. It's awesome on the bed because instead of being "sheety", it's got those rubber parts so one doesn't fold it under the mattress to keep it in place but rather skewers it or maybe embraces it onto it.
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
since there's no escaping it.
From Wikipedia...
a fitted sheet has its four corners, and sometimes two or four sides, fitted with elastic, to be used only as a bottom sheet.
You still make the bed regardless of whether it has a fitted sheet. But if it does, people often call the other one1 a top sheet to distinguish it from the elasticated one (a fitted sheet doesn't work on top).
1 If indeed there is another sheet - increasingly less likely as we move from top sheets under "itchy" blankets to removable covers on duvets.
I'm going to have to make a trip to IKEA because it sounds like you are not describing a traditional fitted sheet. It sounds like you are describing a modified standard sheet where the corners have rubber (for grip) pockets sewn in such that the corners of the mattress fit into the rubber pockets of the sheet which holds the sheet firmly attached to the mattress like a fitted sheet. But this kind of sheet seems like it would be much easier to fold than a traditional fitted sheet. I would still say, "making the bed" with this kind of sheet, but to describe the specific action of hooking the sheet pocket around the mattress corner I would probably say I was, "buttoning" the sheet down, or "dressing" the mattress up.
The name "Fitted Sheet" has already been given.
I was going to suggest "evil sheet of difficult foldiness and doom", but since I see that issue has been dealt with, I'll suggest instead contour sheet as a relatively uncommon synonym used -- so far as I can tell -- mostly by marketers trying to make them sound fancier.
I am interested by your description of them as "fancy" sheets. Growing up in the UK, I found beds with fitted sheets to be frowned upon as "not as good".
I think, though, that this might be more a factor of the amount of extra work needed to provide crisp linen sheets carefully tucked into the corners of the bed. So upmarket hotels generally provide tucked-in apple-pie beds because it provides the impression of hand-cared service, but likely also because there's no elastic to go out on them if you wash them very hot, so they last longer, and are probably cheaper.
