Third street is a casino term most often used to refer to the first betting round in poker variation called Seven Stud Poker.
If you are somewhat familiar with the rules of Seven Stud Poker, the term third street is self-explanatory. During the first betting round, players have three cards in their hands, which is how the term got its name. In addition, there are also the fourth, fifth, sixth, and seventh street and all of them have the same meaning — they refer to the number of cards a player has in their hand.
Stud poker players often use these terms when trying to explain during which stage an action took place. For example, they often say they have raised or folded on the third or sixth street.