Théodred is a rather unfortunate character – his main raison d’être is to die, leaving Éomer as heir to the Riddermark.

Théodred in the book
In the book, Théodred is dead before Gandalf and the others even reach Meduseld, and thus he is only discussed – in the past tense.

He died in a battle at the Fords of Isen (the battle we see Éomer carrying him from in the extended edition of the film), news of his death reaching King Théoden five days before Gandalf arrived in Edoras.

He is then mentioned twice:
Wormtongue: “Théodred your son was slain upon the West Marches: your right-hand, Second Marshal of the Mark.”

“Behold! I go forth, and it seems like to be my last riding,” said Théoden. “I have no child. Théodred my son is slain. I name Éomer my sister-son to be my heir.”

And thus ends Théodred’s role in the book.

Théodred in the film

In the film, Théodred is actually seen alive. His first appearance is at the Fords of Isen, where he lies gravely injured. A group of Rohirrim, led by Éomer, arrive and Éomer orders his men to search for the king’s son.
Théodred is found, still alive, and taken back to Meduseld, riding unconscious in front of Éomer.

In Meduseld, he is then seen lying in his room, Éomer at his side. When Éowyn comes in, she sits down on his bed and softly says his name. She then takes a look under the bandages on his abdomen and grimaces.

Éomer and Éowyn are then seen with Théoden, telling him that his son is gravely wounded. In the next scene in Meduseld, Éowyn tells him that Théodred is dead. After Théoden is released from Saruman’s influence and expels Gríma from his Hall, he looks round and asks where Théodred is.

Straight after that, Théodred’s funeral is seen, his body brought to the barrows, carried on the shoulders of Rohirrim. The townspeople are gathered, making a slow procession behind the body. Éowyn is seen waiting at the open tomb with the other women. As Théodred’s body is passed through hands into the tomb, Eowyn sings in Rohirric:
“An evil death has set forth the noble warrior
A song shall sing sorrowing minstrels
in Meduseld that he is no more,
to his lord dearest and kinsmen most belover.
An evil death…”

The tomb is then shut, and outside, Théoden is seen holding a simblemynë. Then the last mention of Théodred comes from Gandalf: “He was strong in life, his spirit will find it’s way to the halls of your fathers. Westu hál. Ferðu, Théodred, Ferðu. (Be-thou well. Go-thou, Théodred, go-thou.)”