Bullseye

Bullseye is Woody's horse. Despite the fact that he is a horse, he can remind you of a puppy by the way he acts at times. He is extremely affectionate towards Woody, Jessie, and the other toys. He doesn't have a speaking voice, instead he neighs and makes other horse noises.

Toy Story 2
In Toy Story 2, he is a toy horse that is part of Al's Roundup collection, which he intends to sell at a toy museum in Japan. He is the first of the toys that Woody meets when he arrives at Al's Penthouse. Being in a storage for several years, Bullseye is excited to see Woody finally join the collection. He is shown to loathe fights, as seen when he hides into a can when Jessie is about to pounce on Woody for framing her about the TV incident that prevented Woody from getting his severed arm back himself. However, Stinky Pete the Prospector stops the fight, and Bullseye expresses disappointment as much as Jessie does when Woody declares he will leave as soon as his arm is fixed. Later, when Woody is about to leave the apartment after having his arm reattached, Bullseye is still shown to be upset, but later brightens up after Woody, having learned about Jessie's tragic past, decides to stay with the gang to go to the museum.

When Andy's toys, led by an Utility Belt Buzz, break into Al's Penthouse, Slinky Dog and Bullseye have a growling match that Jessie tries to stop, but Slinky Dog coils her up along with Bullseye. After Woody decides to go back to Andy's house, he decides to take the Roundup collection with him, and when he asks Bullseye, the horse eagerly licks Woody. However, Stinky Pete locks Woody, Jessie and Bullseye inside the room, and the Roundup toys end up being loaded into Al's case for its shipment to Japan from Tri-County International Airport.

At the airport, Bullseye manages to escape the case just before it reaches the end of a conveyor belt, but Jessie still remains trapped inside the case as it is loaded into the luggage transporter. Woody and Buzz Lightyear mount the newly freed Bullseye, and they gallop across the airfield to rescue Jessie. Despite being separated from Woody, Buzz commanders Bullseye to follow him, as they are later seen galloping next to the front tires of the plane when Woody and Jessie climb out of the aircraft. Woody and Jessie swing between the tires via Woody's pull-string, and they land on Bullseye, right behind Buzz, just before the plane takes off. Back at Andy's house, Bullseye (and Jessie) is now part of Andy's toys, and Andy has printed each letter of his name onto each of Bullseye's hooves. Finally, Bullseye joins the other toys as they gather round to see Wheezy sing You've Got a Friend in Me.

Toy Story 3
In Toy Story 3, he is first seen in Andy's realistic play scenario when he ferries Woody and Jessie as they chase after a train, which has been robbed by an evil, nefarious bandit couple One-Eyed Bart and Betty (portrayed by the Potato Heads).

Years later, when an adult Andy opens his toybox to decide what he will do with the toys, he puts Bullseye in the bag destined for the attic, but Andy's mom mistakenly thinks the bag as trash and puts it onto the street, but Bullseye, along with the other toys that have ended up in the bag, manage to escape and board the box bound for Sunnyside Daycare Center. Bullseye decides to follow Woody back home when Woody decides to leave Sunnyside, but Woody persistently insisted Bullseye to stay with the other toys as he didn't want him to be lonely in the attic, leaving Bullseye heartbroken.

When Woody, having learned of the intentions at Sunnyside, returns to the daycare center and reunites with the toys, Bullseye gives Woody his hat back. Bullseye also gives Mr. Potato Head (who has caught up with the toys in a cucumber body) his plastic potato body back when the toys escape Sunnyside.

Finally, Bullseye, along with the other toys, is given to Bonnie by Andy. In the end credits, he turns on the boom box to play "Hay Un Amigo En Mi," the Spanish version of the song "You've Got a Friend in Me" to which Buzz and Jessie perform a paso doble.

Toy Description
From Official Website: "Bullseye is Woody's trusty toy steed from the "Woody's Roundup" gang. Bullseye can ride like the wind and leap across giant canyons in a single bound, especially when Woody is in the saddle."

Trivia

 * Bullseye was originally going to have a speaking voice, but John Lasseter decided to make him a non speaking character since he would be more appealing to audiences.
 * During the "Rescue Jessie" scene in Toy Story 2, Bullseye is somehow able to run just as fast as a plane getting ready to take off, similar to how RC can catch up to a moving truck in the first film.
 * Even though he is Woody's horse, Bullseye seems to be more attached to Jessie, as can be seen mostly in Toy Story 3. He can be often seen with her at his side, usually with an arm around him, at other times (such as when something dramatic is going on), he can be seen hiding behind her, almost as if it were for protection or out of habit. This is probably due to spending more time with Jessie while still living in Al's apartment.
 * He also appears in Toy Story 3: The Video Game.