Draw a table listing down the 9 teams in the first row and first column. Cross out the matches between the same team and the repeated matches and you will get the answer:)

Start with Team 1. It will play against teams 2,3,4,5,6,7,8,9 (total 8 games). Continue to team 2. Since it has already played with team 1, it will now match with teams 3,4,5,6,7,8,9 (total 7 games). If you can see the pattern, you can derive the answer quickly without needing to list all the matches (though it is recommended so you don't make careless mistakes). Therefore, the total number of matches will be 8+7+6+5+4+3+2+1+0=36!