This informal idiom is highly popular in both social and professional situations and can mean to annoy someone to the point that they show displeasure towards you. This idiom is usually used in relation to somebody in a position of power or authority, or who is considered dangerous in some way. It would be rare to hear a native English speaker use the opposite term 'somebody’s right side'.
I think I got on the wrong side of Sarah at the party because of my boorish behaviour. Hopefully she will accept my apology and not hold it against me.
Mike is someone with a lot of power and influence in the industry. You really don't want to get on the wrong side of him.