What happens if both the dealer and the player got the same value in blackjack?

If both the dealer and the player have a Blackjack it is a push.

However, it is possible for a player to win the hand under certain conditions. If the dealer is showing an Ace the player can insure his bet. If the player makes the insurance bet and the dealer has a Blackjack, the hand is a push and the insurance bet is a win. This is commonly know as "even money".