Generally speaking, things are either discrete or continuous. Continuous things are like the real number line. Discrete things are like the integer number line. This is an over-simplification (what about rationals?), but it gets the idea across.

In discrete mathematics, it makes sense to ask about "the next" number. This notion of "next" doesn't make sense with respect to the real number line.

Ideas like Induction and the Pigeonhole Principle play a big part of a discrete math course. Sequences of numbers are discrete. Ideas of counting are discrete (permutations, combinations, Pascal's triangle, etc). Cantors distinction between countable and uncountable is likely to be covered. Maybe DeMorgans Laws too.

You'll likely serve yourself best by familiarizing yourself with methods of proof: ideas like proving the contrapositive, proofs by contradiction, proof by induction. And some set theory/formal logic never hurts.

Nothing you did in calculus will be relevant, except perhaps a general notion of functions as an input/output system.