What is this game called?
The game you describe seems to be a fairly typical example of the Shithead / Palace / Shed / Karma / China Hand / Ten-Two Slide (Wikipedia) family of card games.  These games are known by a variety of names around the world, and have many local variations.  The game you describe does, however, contain most of the basic elements:


Players are dealt three initial sets of cards: one set in their hand, one face up on the table, and one face down.
The goal of the game is to get rid of all your cards, both those in your hand and those on the table, by placing them on top of a common discard pile.
You must first play the card in your hand, then the face-up cards from the table, and finally the face-down cards, which must be played blind without looking at them.
On each turn, you may place one or more cards of the same rank onto the pile, but only (with some exceptions) if they equal or exceed the rank of the current topmost card on the pile.
The most common exceptions are that a 2 or a 10 may be played at any time; a 2 allows the next player to play any card, while a 10 (or, commonly, any four cards of the same rank together) will clear the whole pile and allow you to restart it from any rank you want.
If you cannot play any card (or fail to play a blind card), you must pick up the entire discard pile into your hand and let the next player restart a new pile.