This is not going to be a complete Udyr guide, only how I play him. Udyr is an immensely versatile champion that is unusual has he does not have an ultimate.

In this I will cover the reasoning behind my Runes, Summoner Spells, and Mastery choices, my preferred Item list, my skill sequence, and lastly an idea of how to Jungle with him.

This build is based around Udyr jungling and as such don't use it for Udyr in a lane, and since your reading this.... dont lane Udyr, ever. There are far better choices for a lane.

Lastly its important to acknowledge one critical flaw in Udyr, he is not a good ganker. He is however top of his class at everything else. His ganking ability is limited to a single limited speed boost ability with a short stun at the end of it. With support he can be a terrifying ganker, but without it he has to rely upon superior positioning, and the opposing team not warding properly.

The theory behind Summoner Spells for Udyr is very simple. All but very few junglers require Smite to be able to engage and survive the three key neutral minions; Blue Golem, Red, Mini Golems. This is the must select Summoner Spell.

The other key Summoner Spell is Flash. Flash will let you get into places to get kills that would be otherwise impossible, and get out of situations that would be lethal. This is especially true when you are in the jungles as you can flash over walls to escape.

The last viable, but not recommended, summoner spell is Ghost. Ghost will let you get out of situations quickly, but isnt considered as useful due to the insane speed boost that Blazing Stampede will give Udyr. Combing Ghostwith Blazing Stampede will lose around 50% of the effectiveness due to the soft cap for movement speed.

UPDATE

I'm now considering Ghost as a better option to Flash. While Flash will allow you to escape by jumping walls there is an advantage to Ghost. Ghost allows you to mitigate the one early game weakness of Udyr in his difficulty to effectively gank. Ghost will allow you to close range and prevent enemies from fleeing allowing you to increase your ability to gank.

I believe there are only two valid mastery builds for Udyr. Either maximize your offence, or maximize your jungling and survivability.

Recommended Build: Survivability

Alternative Build: Offense

Runes for Udyr is very much personal choice, there are so many options due to how Udyrs abilities work.

My preference is to maximize his speed in movement and attack. Combined with early abilities like Wingborne Storm the attack speed increase allows for significantly more of engulfing flame effects creating more damage and removing minions faster. Late game it stacks well with [Tiger Stance] giving insane attack speed.

Alternate builds pick runes to maximize ability damage for Wingborne Storm or attack damage for Wilding Claw. I think each of these is valid, but only a boost for early, or late game. Attack Speed and Movement Speed bonuses are consistently useful. Damage reduction is less of an issue with your ability to escape with Blazing Stampede and take damage with Iron Mantle.

Movement speed should be self explanatory for Udyr, faster movement between neutral minions in jungles, and faster movement to catch out enemy champions for quick kills.

This is a default pattern that should be adjusted based on the composition of your opposing team. If you are running into trouble with Attack Damage based champions look to buy Thornmail or if you are getting attacked by Ability Damage champions look to get Banshee's Veil. Never follow any guides item build blindly. Understand the reasoning and apply your own situation to it.

Starting Items

The starting items for any jungler are fairly static, you need to have armor to handle the damage output from minions and potions to avoid having to return and heal for as long as possible.

Early Game Items

I recommend your first purchase when returning is Rejuvenation Bead. This will improve the length of time you can remain jungling and help with your ability to both jungle and have enough health to engage enemy champions in lane, or counter-junglers.

[[Boots of Speed], [Null-Magic Mantle], and [Ruby Crystal] are fairly interchangeable as your next purchases. This is merely my preferred order.

Next return you should try to complete both Mercury's Treads size=55 and Heart of Gold.

Mid Game

The mid game focus of this build is massively boosting your attack speed, as well as your ability to survive and recover from ganking, and failed attempts.
Wit's End is a stable for Udyr builds and combines Attack Speed, Magic Resistance, and boots the damage you are applying each hit by 43 Magic Damage. This is dangerous as your Attack Speed builds and you move into using Wilding Claw to deliver your damage. Zeal Brings in more attack speed and increases your movement speed to the point where you should be reaching the soft cap when using Blazing Stampede. Warden's Mail improves your survivability and recovery time and can give you a boost when ganking if the opposing champion turns to fight.

Late Game

Late game time, we need to get into team battles and take down those last towers. To do so we need to continue to build our attack speed, hit like a Mack Truck, and be able to survive it.
To do this we bring in key items. Randuin's Omen Boosts your survivability for those team fights as you will likely be initiating with Blazing Stampede. Phage reduces the opponents chance to run away and give you more time to get off those lighting fast attacks. Trinity Force is just good for everything with a great passive. Warmog's Armor give you a big health boost to ensure survival and stacks with our last item to cause massive damage. Lastly Atma's Impaler takes all that health you have built up and turns it into damage. Combined with your Attack Speed boosting items, Wilding Claw, and your ability to stun a champion coming in with Blazing Stampede many champions can be killed before they can react.

Other Considerations

Don't forget the importance of Sight Ward, Vision Ward, and Oracle's Elixir. These three items are critical to having zone control and being able to jungle effectively. These become even more valuable as you push into the opponents jungle to take blue and red buffs.

I've combined these as I think they all chain in together. As I have mentioned before, Udyr is different in that he has no ultimate, and he has very very low cooldowns. He is also different as all his abilities are available at level 1.

Abilities

Monkey's Agility A underlooked passive that builds rapidly to become very lethal with the speed you can change stances. It caps at 3 stacks totaling 30% Attack Speed.
Tiger Stance This skill is best used as a late game skill when your item bonuses have made it lethal. This needs to be activated as often as possible as the high burst damage is the most effective part of this stance. It will however drain mana quickly and is unsuitable for early game.
Turtle Stance This is an amazing ability for two reasons, the activation shield is affected by resistances, and the shield will persist after you change stances for a limited time. This will be a staple ability for jungling, and for absorbing damage on team fights.
Bear Stance This is your lead in skill for ganking. It will allow you to catch and stun opposing champions. It has utility for fleeing, adding stuns to minions, and interruptions. This ability does experience diminishing returns on its speed boost when you hit the soft cap.
Phoenix Stance This stance is different to the others as it maintains its usefulness after the initial activation. This is your early game superpower and will be your highest damage tool until late game when Tiger Stance starts to be more effective.
Based on this we want to level Wingborne Storm and Iron Mantle quickly with minimal points in Blazing Stampede for early ganking. Late game we want to focus on Wilding Claw to boost single target damage for towers and champions.

Ability Use

We have two ability sequence types; Jungling and Ganking.

Early Game Jungle Sequence Iron Mantle> Wingborne Storm
Mid/Late Game Jungle Sequence Iron Mantle> Wingborne Storm> Blazing Stampede> Iron Mantle> Pheonix Stance

Early/Mid Gank Sequence Blazing Stampede> Wingborne Storm> Iron Mantle> Wingborne Storm
Late Game Gank Sequence Blazing Stampede> Wilding Claw> Iron Mantle> Wilding Claw

In my opinion Udyr is the most versatile fighter in the game. He is one of the strongest junglers even though he takes a little longer to build to critical mass.

Jungling requires team support, and that goes double for counter jungling. To familiarise yourself with the actual process for jungling check out this guide Jungling 101, and Jungling 202. Combine that, with this build and watch the world fall to ash around you.