Theódred was the only child and son of Théoden and Elfhild, who died when her son was born and thus Théodred was raised alone by Théoden. Tolkien doesn't really write about these times and thus we can only assume about the youth that Théodred had.
What is known is that Théodred was an energetic man who was loyal to his father and that he did all he could to counter Wormtongue when he got influence over the king. Because of his efforts Saruman had a difficult time to conquer Rohan.
As the son of the King Theódred was appointed to be the Second Marshal of the Mark, having the region of the Westmark and his base at Helm's Deep. During the period were Théoden came under the illness caused by Gríma, Théodred, without orders from his father and probably, if he had known, against his will at this time, took general command of the army and he thus acted like he was the King. Theódred assigned Elfhelm to guard Edoras while he himself went to battle when he heard about the upcoming invasion of Saruman in the Westfold. But he didn't know about the full strength of Saruman's forces and was surrounded at the Fords of the Isen by Uruk-Hai, Orcs and Wargriders where he was eventually killed by a large Orc. This happened five days before Gandalf, Aragorn, Legolas and Gimli arrived at Edoras. He was 43 years old at the time of his death.
More about the Battle at the Fords of the Isen here.
Théodred was also in the movie of Lord of the Rings: The Two Towers. There were significant changes to his character. In the Extended Edition we see how Éomer rides to the Fords of the Isen to find Théodred dying and he brings him back to Edoras, where the son of the King eventually dies and is buried in a burial mount.
He is a lot younger in the movie, in the beginning of his twenties I assume and so younger than Éomer who is 28. This is all for the dramatic effect in the movie and that's why Peter Jackson put Théodred in the movie.
Théodred is played by Paris Howe Strewe.