"Regarding" significa "en cuanto a". You are saying "as for X", where X is the thing you are talking about. Por ejemplo:

   Regarding your school report, it was very well written.*

   or

   What is this regarding?

"Regardless" significa "a pesar de todo"/"pase lo que pase"/"en cualquier caso". Ejemplo:

   I'm going going to movies regardless of what my parents say. (No matter what my parents say, I will do it anyway.)

  or

  There may be obstacles, but we will succeed regardless.

I hope that helps

*Regarding is often used to switch topics when discussing more than one thing.