Getting good ranks in Dustforce is an integral part of the game. In order to see more content, you must obtain more keys through getting high scores, and to see all of the content in the game you must get SS ranks on all the levels.

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This should be self-explanatory. Completion is how much dust you can clean before completing the level. This includes:

• Dust-coated surfaces. Dust created on surfaces by heavy attacks on enemies WILL count towards this
  
• Dust blocks
  
• Every kind of enemy, including the docile ones (balls of leaves, small gargoyles, etc)

AFAIK, not cleaning dust directly subtracts from your Completion grade up to a minimum of D rank.

This one isn't as straightforward as Completion, but shouldn't be difficult to understand.

You start levels at S rank in Finesse, and each time you drop a combo your rank lowers down a grade. This is typically caused by getting hit by enemies, falling into pits, touching spikes, etc., but not having a combo at all means you can do those and still get a good Finesse score.

Note that going slowly is NOT directly counted against your Finesse rank. If it's more comfortable to you, take your time and go slowly as long as doing so doesn't cause the combo to be dropped.

If you want to score S in Finesse, it helps to disable checkpoints from the options menu. That way, falling into pits or touching spikes automatically sets you up for a fresh run.