a is a noun meaning happiness, so nina furaha literally means 'I have happiness'
i is a verb meaning to rejoice or to be happy.
Good answer! I've never seen this particular resource on DuoLingo, so that's something I'll have to begin looking at. From the example they give, I need to wonder whether you may only use the verb form when you're telling what you're happy about. It might be that when you are simply announcing that you're happy, you need to express it as "having happiness". I don't know, but I surmise:)