Eight members of a chess club are having a tournament. in the first round every player will play a chess game against every other player. how many games will be in the first tournament

Same as handshake problem.
Each of n persons plays against n-1 persons for a total of n(n-1)/2 games.
The division by 2 is necessary to avoid double counting the game A vs B and B vs A.

For n=8, then number of games played in the first round is 8(8-1)/2=28 games.