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If they're both countably infinite, eg two infinitely long lines of men and women, then you can have equal pairings, any finite number of leftovers, or an infinite number of leftovers (of either gender you choose).
If we use the "two single-file lines" model, a perfect matching would be matching people 1-to-1 as they come off the end of the line; a finite number of leftovers of men/women would be taking that many people off the front of the line, then matching 1-to-1 the remainder of both lines; and an infinite number of leftovers would be taking 2 from one line, making one of them leftover and matching the other to someone in the other line.
These correspond to mappings x -> x for N to N, x -> x-n from N\{1..n} to N, or x -> x/2 for 2N to N (sort of, I'm abusing notation).
Of course, in "real life" these procedures would never terminate. Luckily, infinite amounts of people don't exist in real life either.
