No Limit Hold'em is a variation of Texas Hold'em poker where there is no ceiling placed on the betting. Any active player can bet any amount at any time in the hand, provided it is their turn to bet. This twist makes no limit Texas Hold'em a very different beast to limit Hold'em.
At any point in the hand you face the potential to have to stake all your chips in response to an opponent's bet. This means that you have to be far more confident of the hands you play. Texas Hold'em poker can be traced back to Dallas Texas in 1925, according to Wikipedia. One can speculate on when the no limit Hold'em variation was invented, so to speak, but for the sake of convenience one can take it's introduction to the poker rooms in Las Vegas as the pivotal moment in its development into the most popular form of poker on the planet today. It is now the most popular form of online poker.
No limit Hold'em was introduced to Las Vegas in the late 1960s by a group road gamblers - professional poker players who made their living travelling from town to town, pursuing the most profitable private poker games. They had to deal with crooked games and players, armed robbers raiding poker games, and the forces of law enforcement, as these games were illegal. This group included some of the legends of the game, like Johnny Moss, Amarillo Slim Preston and Puggy Pearson. Their arrival in Las Vegas, and Binion's establishment of at poker room at Binion's Casino in 1970, triggered the modern era of no limit Texas Hold'em. In Binion's held what became the first World Series of Poker, where after a few days play, the players voted Johnny Moss the winner.
Each subsequent year saw the tournament grow, slowly at first, until Binion managed to get it televised in the 'eighties. The advent on online poker rooms and breakthroughs in camera technology have seen it rocket even further in popularity in the new millennium.