Live here uses a pretty standard swiss pairing algorithm. For the first round, players are listed in rating order, top half of the field, plays the bottom half.
With 10 players
1 plays 6
2 plays 7
3 plays 8 .....
After each round, the same process is done, but it is done by score groups. So, after round 1 all players with 1 point are their own group, all players with 0, etc, and the same method from above is used. There are methods to handle odd number point groups, so some players will get paired with someone in a different score group. You also have to worry about color allocations and making sure two players don't face each other more than once.
How can I calculate which opponent I get?in tournament?