For ten players A-J. Just keep A constant and rotate the other nine letters, starting with them listed down the left-hand column and back up the right-hand column (see appendix for detail).   The numbers represent rounds and the letters represent the players.
 

For a chess tournament where the first-named player has the white pieces, in order to ensure that each player has a colour distribution as even as possible, player A would alternate from the left-hand column to the right-hand column, and the nine rounds listed above would be played in (for example) the order 1, 6, 2, 7, 3, 8, 4, 9, 5.

The same system works for any even number of players.  For an odd number of players one of the letters is designated as the "bye".

If the players are listed as A, B, C ... in order of strength, the above draw has the top two players meeting each other in the final round, which is a good thing.  Many people do not realise the important of seeding an all-play-all (or "American") event to ensure that the top players do not meet in early rounds.  If they meet in early rounds it destroys further interest in the event because once they have played each other the almost certain winner will be apparent; and it also tends to decrease the number of spectators and increase the number of forfeits and withdrawals.

Appendix

More detail on creating the table for an even number of players. In this example the players are numbered 1-10

1. Allocate each player a number, then list the players in pairs with the numbers going down the left-hand column and back up the right-hand column. For 10 players there will be 5 rows as follows: 1-10 2-9 3-8 4-7 5-6. This represents the draw for round 1.
 

2. For round 2 leave number 10 unchanged and add one on to each of the other numbers, but you can have only one number 10, so when you add one onto 9 call it 1, not 10. This gives:

3. For round 3 repeat the same process, leaving 10 unchanged and adding one onto each of the other nine numbers, except that 9+1=1, not 10.