Because the first two players to the left have already put in chips, the first player to act pre-flop is the third player to the left of the dealer. Now that every player has hole cards, there is a round of betting. This doesn’t mean Texas Hold'em never uses antes: some games have antes just to make it more exciting, and late into tournament play, antes might be introduced to keep the game moving. During tournaments, blinds raise incrementally to keep the game moving. The first player to the left puts in the small blind, usually one-half of the minimum bet the next player puts in the big blind, usually the size of the minimum bet. Most Texas Hold'em games are blind games: the players to the left of the dealer each put in a small number of chips before any cards are dealt. This is the most popular way to play, but not the only one: players can play limit (they bet a set amount, and there are only so many raises allowed during each round of betting) and pot-limit (the maximum allowable bet is to match the number of chips already in the pot).
Texas Hold'em is usually no-limit: at any moment, a player can bet all of their chips. But Texas Hold'em is also an incredibly popular cash game: players buy as many chips as they have cash, and come and go as they please. Texas Hold'em is a popular tournament game. In tournaments, every player is given the same number of chips, and they play until one player wins them all.
