Rank scoring means that for each entry on a single game, a player is ranked + scored by comparing their score to the others on the same machine and assigned some sort of point value.

For example, let's assume you have a Fish Tales in a tournament:
Larry got 14,567,908
Curly got 1,563,697
Moe got 9,456,287
Shemp got 18,999,321

Fish Tales ranking:
1st - Shemp
2nd - Larry
3rd - Moe
4th - Curly

Now, from this ranking, you need to assign some sort of point value. This is up to the tournament director's discretion, but it usually is done with (100-90-85-84...) scoring. Players below rank 87 get zero points for their score.

So using that scoring scale:
Shemp has 100pts
Larry has 90pts
Moe has 85pts
Curly has 84pts

If you do this for each machine in the tournament and add up the final points from each machine, you can determine, overall, who the top players are in that tournament round.

Note: There is a tournament program that automates this (google for Scott Danesi's Arcade Tournament Manager, the PAPA Drains Tournament Manager supports this too, although I don't think this open to the general public yet), or if you have spreadsheet skills you can do it in Excel or LibreOffice. If you want the scoring algorithm, I've written a custom function for LibreOffice that implements (100-90-85-84-...) scoring.