Question

How can functional groups be polar?

Answer 1

Polarity isn't determined by the 'type of molecule' meaning functional group or not. Instead, it's about the types of bonds between atoms.

When 2 equally strong (electronegative) atoms are bound, the sharing of electrons will be equal between them.
However, when a bond exists between a strong and weak atom, the stronger atom will take advantage of the weaker one and 'hog' the electrons. This makes the stronger atom slightly more negative, and the weaker one slightly more positive

If a functional group is composed of an atom that has strong-weak bonds, the group will be polar.
Examples include:
Alcohol - bond between O and H
Amine - Bond between N and H
Halogens - bond between X and the parent chain (X refers to F, Cl, Br, I)
Thiol - bond between S and H

These are just a few to give you an idea. For a detailed review of functional groups, watch the video below:

Good explanation, nice figures on this website: http://www.whatischemistry.unina.it/en/hphil.html