'Much Ado About Nothing' is considered a comedy because it all ends happily ever after. There are no 'good' characters (or protagonists) left dead or unhappy, only the evil ones (that is, Don John, Borachio and Conrad), who's plans are left unfinished and ruined.
The play is also considered a comedy because of the humorous interaction between characters, such as the witty insult exchanges between Benedick and Beatrice in Act 1 Scene 1, and the slapstick comedy of Dogberry and Verges at the guard. These characters are characterised in a way so that the audience can feel an immediate connection to them - they are the clowns of the play, who make us laugh and enjoy ourselves in light of all the tragedy with Hero and Claudio. As well as this, they also (conveniently) save the day by arresting the antagonists.