Ulfric’s War Axe is less a weapon than a message. Ulfric Stormcloak hands it to you to deliver to Jarl Balgruuf – a blunt way of asking whether Whiterun is with him or against him.
How to get it
Ulfric gives it to you if you join the Stormcloaks, as part of Message to Whiterun. Delivering it to Balgruuf moves the civil war on to Battle for Whiterun. (Balgruuf only makes his decision after you’ve finished Bleak Falls Barrow.)
Keeping it
As a quest item it can’t be dropped, but it can go on a weapon rack in your house. With the Misdirection perk you can also pickpocket it from Ulfric.
Stats
A steel war axe: 10 damage, 12 weight. Improved with a Steel Ingot.