Special moves use Spc Atk, and Spc Def in their damage calculations.
Normal moves use Atk, and Def in their damage calculations.
How do I know which move is which?
How to tell if a move is special or physical
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Best Answer
When you go to the move description, physical moves will have an orange and yellow "explosion" symbol, whereas special moves will have a purple "swirl" symbol in the same spot.
Images courtesy of Serebii.
As for moves your opponents' pokemon are using against you, you'll either have to look it up in an online database or wait until one of your own pokemon learns the move. There is no way to check that.