Mad-Eye had a staff which he used to cast a spell on a Death Eater in HP5.
Is this staff his wand (or does he not have a wand) and if that was his 'wand', how did it function?
Mad-Eye Moody had a wand, but his staff could have served as a backup wand.
Mad-Eye Moody had a regular wand as well as the staff.
He only uses the staff to cast a spell once (in the movie). Other than that, he uses a wand, and his staff is never said to have magical capability in the books. For the most part, his staff is basically just an ordinary walking stick,even if it has the ability to be used to cast spells.
It's possible that his staff may have had some kind of ability to function as a backup wand, in case he was ever disarmed in a duel, because Moody believes in constantly being prepared and ready for any situation. Ollivander has confirmed that magic can be channeled through almost anything, although well-matched wands are the most effective choice.
"If you are any wizard at all you will be able to channel your magic through almost any instrument. The best results, however, must always come where there is the strongest affinity between wizard and wand. These connections are complex. An initial attraction, and then a mutual quest for experience, the wand learning from the wizard, the wizard from the wand."
- Harry Potter and the Deathly Hallows, Chapter 24 (The Wandmaker)
It seems to be made of wood, like most wands are, so it could conceivably function as a very oversized wand, especially if it was made from a wood with magical properties like most common wand woods. It probably wouldn't be as good as a small wand, which would be much easier to wield, but it could work somewhat.