Puzzle was a talking donkey, and his name suited him very well. He was rather dense, but he had a good heart and did what was right in the end.
He was a "friend" of Shift, who was a conniving ape that abused Puzzle's trust for his own gain. Puzzle was often tricked into doing whatever Shift told him to do, even if he knew it was wrong.
When Shift discovered the skin of a lion in Caldron Pool, he had Puzzle retrieve it at great peril and later wear it. The loyal donkey originally didn't want to wear the skin, but nevertheless was tricked into imitating Aslan, and Shift used him to gain control over his fellow Narnians, and later hand them over to the Calormenes.
The skin wasn't even that good, mainly because of Puzzle's face coming out of the mouth of the skin. That's why Shift only brought him out at night and didn't make him come too close to anyone, or make brays or hoof noises. Anyone who had seen a real lion, let alone Aslan, before wouldn't be fooled, but no one in Narnia had in years.
Puzzle was rescued by Jill Pole, who made him see the error of his ways. She protected him from Tirian who at first wanted to kill him for pretending to be Aslan. Puzzle then joined the group of good Narnians in the Last Battle of Narnia.
When the real Aslan arrived in Narnia, he had some private words with Puzzle that made his ears go down, and some more words that made his ears go up. After this private discussion, he was officially pardoned and led into Aslan's Country. Without the lion skin, everyone thought Puzzle looked beautiful with his coat and face.