Definition of variance
The variance in poker can be understood as an hourly rate. A winning player, who would win $50/hour on average, may, in a 4 hour session, have won $100 the first hour, lost $50 the next hour, won $200 the next hour and lost $50 in the 4th hour. This is only one of the possibilities. He could very well have won $100 every hour or lost $100 every hour. The difference between his actual hourly rate (his hourly rate based on thousands of hours of play) and his hourly rate over the short term, is called variance.
The variance has many faces
Randomness in Texas Hold'em is present in many aspects of the game. For example, in each hand, each player is dealt 2 hole cards. These two cards are dealt at random. The player has no control over the choice of these cards; the player receives the two cards that the dealer has randomly dealt to him/her. The decisions are made by the dealer after receiving the cards. For example, the player can decide whether or not to play this hand. How he plays his hand (if he plays it) is also at his discretion. He will receive a pair of aces, for example, 1 time out of 221 (on average of course). Of course, out of 221 hands, he may receive 3 or more times a pair of aces if he is "lucky" and none if he is "unlucky". These are some of the many faces of the variance.