How did it happen in the movie?
The beacons section in the film has three parts – Gandalf telling Pippin to go and light the beacons; the beautiful sequence of photography showing the beacons forming a chain of signals all the way to Rohan; and Aragorn bringing news of the beacon lights to Théoden.

Pippin’s task — Gandalf informs Pippin that he has a task for him – which would be another opportunity for one of the Shire-folk to prove their great worth. As soon as he tells Pippin that the task is to sneakily light the beacon fire, Pippin runs off.

Beacon sequence — Pippin lighting the beacon fire occurs silently – he climbs to the top of the beacon, and with some effort, dumps some oil on the wood, starting the beacon fire. He quickly realizes it’s not particularly clever to stay on a lit pile of wood, and begins climbing back down. Only as he is climbing down away from the beacon do the soldiers guarding the beacon realize that it is lit.

Then a sequence of aerial photography shows all of the beacons between Gondor and Rohan being lit, one by one. This is an exceptionally pretty part of the film, and shows off New Zealand scenery wonderfully! There’s only one small problem – when choosing which peaks to use for the fires, Peter Jackson seems to have forgotten that people would actually have to be able to get to the beacons to light them (and I don’t mean by helicopter), and also that the beacon guards would have to have at least some place to shelter, if not live in semi-luxury, during their stints in the middle of nowhere, constantly looking out for a signal that is very unlikely to ever be seen.

The lighting of the beacons was seen as a glimmer of hope in the darkness falling over Minas Tirith and its soldiers:
Gandalf: (quietly) Amon Dîn.
Guard: The beacon! The beacon of Amon Dîn is lit!
(Denethor peers out a window and sneers.)
Gandalf: Hope is kindled.

Théoden’s decision –- In both book and film, Théoden’s decision is immediate when the summons to arms comes from Gondor. In the film, it is Aragorn who brings him the news of the beacon fires:
Aragorn: The beacons of Minas Tirith! The beacons are lit! Gondor calls for aid!
(Théoden looks up from a map. Éowyn runs to stand next to Éomer. Everyone watches Théoden’s reaction.)
Théoden: (pauses, and then calmly proclaims) And Rohan will answer!
Théoden: Muster the Rohirrim! Assemble the Men at Dunharrow. As many Men as can be found. You have two days. On the third, we ride for Gondor and war.

What happens in the book?
Everything is different in the book. While the beacons appear, they play a very different role. They are not lit as they are in the film, though Gandalf tells Pippin of how the beacons were once used: “Pippin became drowsy again and paid little attention to Gandalf telling him of the customs of Gondor, and how the Lord of the City had beacons built on the tops of outlying hills along both borders of the great range, and maintained posts at these points where fresh horses were always in readiness to bear his errand-riders to Rohan in the North, or to Belfalas in the South. ‘It is long since the beacons of the North were lit,’ he said” [5.I.]

Instead of using the beacons, the plea for aid in the book comes to Rohan in the form of Hirgon carrying the red arrow. Hirgon, an errand-rider of Gondor, came to Théoden at Dunharrow, instead of Edoras. He bore in his hand a single arrow, black-feathered, barbed with steel, and with a painted red tip.

“He sank on one knee and presented the arrow to Théoden. “Hail, Lord of the Rohirrim, friend of Gondor!” he said. “Hirgon I am, errand-rider of Denethor, who bring you this token of war. Gondor is in great need. Often the Rohirrim have aided us, but now the Lord Denethor asks for all your strength and all your speed, lest Gondor fall at last.”
“The Red Arrow!” said Théoden, holding it, as one who receives a summons long expected and yet dreadful when it comes. His hand trembled.” [5.III.]

Immediately he agreed to come to Gondor’s aid, saying he would set out after the weapontake on the morrow, managing to gather the same amount of men as in the film, but in half the time!