https://ffxiv.gamerescape.com/wiki/D...erms_and_Slang

DPS checks refer to points in fights that require you to put out a certain amount of damage under a strict time limit to pass.
A good early example is the "infernal nail" in Ifrit's fights. If you don't destroy it in time, the party will wipe (everyone dies). This is the most basic DPS check.

Rotation refers to the sequence of skills you use in a fight. Specifically, some jobs have different ones at different levels. Using my black mage example from earlier, the "Flare" rotation is the best one you use at level 50, since it let's you use Flare, the most powerful spell a black mage has at that point, but a spell drain drains all MP, as often as possible.

I'd recommend looking up rotations for your DPS or tank of choice.