Sort of. Basically, at the start of each hand, the person to the left of the "dealer" starts off with a small blind. The player to the left of him then follows with a big blind, which is usually twice as much as the small blind. Then betting begins as normal after everyone gets to see their hand of cards. Everyone aside from the people with the blinds have to match the big blind or fold; essentially, the blinds are the same as any other bets for a hand, they're just forced. Of course, raising is allowed during this part. After the hand's done, the "dealer" passes to the left and the blinds follow. (One person is still designated as the dealer for this, even if it's casino staff that's actually doing the dealing.)
The actual game we'd be playing is called Texas hold 'em. Quick overview of how it's played: After the blinds, everyone gets two cards dealt to them, and betting continues with the person to the left of the big blind. After betting is finished, three cards (the "flop") are dealt to the middle of the table, face-up, and another round of betting begins. After that round of betting, a fourth card (the "turn") is dealt to the center of the table, more betting, and then a fifth card (the "river") is dealt. One more round of betting, and then each player still in the hand reveals their pocket cards, with the player able to make the best five-card hand using their two cards and the five community cards wins the hand. (Usually, out of tradition, the dealer "burns" the top card of the deck before dealing the flop, turn, and river; that doesn't impact the game at all though, it mostly just looks fancier and prevents against someone knowing the next card in the deck if it's marked.)
As for resetting everything after someone runs out of chips... hm. Not a bad idea, actually. If that's the case, everyone will have very few starting chips, since otherwise it would be rather easy to simply pass constantly and not have to worry about running out if your only bets are when the blinds get to you, since chances are someone else will run out before that happens. Say, 5 chips to start, small blind of 1 chip, big blind of 2 chips, and the blinds don't change. As the hands go around the table, you'll have to pay the big blind at some point, and the small blind the next hand, and then as soon as the big blind gets to you again you're out of chips.