Do something unconditionally and you do it with absolutely no exceptions or reservations. If you love your cat unconditionally, then you love her even when she ignores you or makes a mess.
The adverb unconditionally is all about conditions — requirements or stipulations. When things are done unconditionally, there are no conditions at all. If you know your grandmother adores you unconditionally, you're certain she'd never stop loving you, no matter what.