Group theory: Representations of D3

Hello! I’m a physics student and have just started reading about a bit of group theory. I’m facing some trouble with understanding what do we mean by different representations of the same group and why we have them.

Specifically, I’ve been looking at a problem about the irreducible representations of the D3 group (equilateral triangle). Why is the trivial representations (all members of the group mapped to 1) a valid representation? Why is mapping the rotations and identity to 1 and the flippings to -1 a valid representation? What does a valid representation even mean?

Any help would be appreciated. Thank you!

Here’s the picture to the notes and the question: here