Example 1
My little sister's soccer team gives every player an MVP trophy at the end of the season regardless of her actual abilities.
How adorable. In this sentence, regardless means without concern, as in every little girl gets a prize without concern for her soccer skills.
Example 2
I know you think it's sweet that Cole sends you love notes in Pig Latin and knitted you a sweater using his own hair. Regardless, I think you should break up with him.
In this example, regardless is used to indicate that the speaker isn't concerned with the nice (and kind of weird) things Cole has done. She still thinks her friend should kick Cole to the curb.
Example 3
Violetta wears sweatpants everyday regardless of weather or occasion.
We love our sweats as much as the next person, but we don't know how practical this is. Sweatpants at a mid-July pool party? Sweatpants at a funeral? In this totally casual example, regardless means that Violetta doesn't take the forecast or where she's going into consideration before she gets decked out in her favorite sweats.