Pure numbers have no dimension: 1, 5.2, 3.14159
(sometimes constants do, however!)
Alternately, sometime all units cancel out: 6 ft/1ft = 6
Dimensionless quantities can be either pure numbers, or dimensionless groups (expressions where all the units cancel).
For one thing, dimensionless groups are useful for "pure" comparisons.
Compare two quantities using a dimensionless group
Which is further away, Cleveland (136 miles) or State College (223 km)?
Must match units in order to compare (in miles or km?).
Dimensionless groups can be any expression whose units cancel! Here we consider ratio of lengths and times, but force, pressure, resistance, etc. all work as well! The only thing you can't have is a ratio of things with different dimensions! (Obviously, since they would no longer cancel.)