Yes, you can fight from horseback without stirrups. The advantage is that they let you impart more force to your melee attacks without the attendant risk of being dismounted, especially for the downward blows needed for dealing with infantry, and make it harder to dismount you as well. An added benefit is that they make it easier to mount and dismount as well, so riders with heavy armor and weapons can do so unaided rather than being helped into and out of the saddle. Stirrups let cavalry deliver much more effective shock attacks, which goes a long way toward making them an effective combat force.
Stirrups also make horse archers slightly more effective because they give the rider a steadier seat, which increases his accuracy, but the advantage here is minimal, compared to their effect on melee attacks.
Stirrups also make horse archers slightly more effective because they give the rider a steadier seat, which increases his accuracy, but the advantage here is minimal, compared to their effect on melee attacks.