Udyr can fill multiple roles on a team, and has the ability to focus entirely on a certain path to create a direct bonus for his team, or to directly counter an enemy. As a pick, he can be the team's jungler and roamer, ganking and holding lanes when needed. Oddly enough, he's also a very solid laner and generally can hold a 2v1 lane himself if played accordingly, though he will struggle against the lane kings such as Heimerdinger and Mordekaiser. As a counter pick, he does well as a soft anti-carry for damage per second enemies due to high innate dodge and high mobility. Udyr can also fulfill the role of a very strong soft carry, dominating the enemy team for most of early and mid game, though his carry potential weakens the longer the game goes on. This can divert focus from a true carry such as Kog'Maw or Tristana allowing them free farm.
Udyr Team Comps
A very balanced team, good damage, crowd control, mobility, push potential, and map control. Udyr fulfills the role of Semi Carry/Jungler while Heimerdinger 2v1's and pushes, which will lead to a very farmed Heimerdinger. Morgana can mid whilst Rammus babysits Yi until Yi hits super carry
Udyr as a Counter Pick
Udyr has a good mix of burst, mobility, and sturdiness; this allows him to close gaps and do high amounts of entering damage. He also has high innate dodge, and all of these elements make him an exceptional anti-carry. You can go straight at enemy carries in fights, as well as prioritize them while ganking from the jungle, and you have all the tools necessary to shut down an enemy DPS.