Learn French: La vieille maison a brûlé cette nuit. - The old house burned down last night.

While you might expect a reflexive or passive pronominal verb for a fire happening to a house, French simply uses the active verb brûler when objects or buildings are consumed by fire. The pronominal verb se brûler is almost exclusively reserved for living beings accidentally burning their bodies (e.g., Je me suis brûlé - I burned myself).

Where is the word for 'down' in this French sentence?

It doesn't exist here! English relies heavily on phrasal verbs ('burn down', 'burn up') to show total destruction. In French, brûler carries the whole meaning on its own. If you want to heavily emphasize the destruction, you add an adverb instead of a preposition: La maison a complètement brûlé.

Wait, doesn't cette nuit normally mean 'tonight'?

It can, but its meaning shifts depending entirely on the verb tense! Cette nuit literally translates to 'this night'. Because the verb a brûlé is in the past, the phrase refers to the night that just ended ('last night'). If the sentence used a future tense, cette nuit would mean 'tonight'.

If I wanted to say someone intentionally burned the house down, how would that look?

In our sentence, brûler is used as an intransitive verb (meaning it takes no direct object) to describe the event of the fire. To imply that someone else set the fire, you would switch to the passive voice: La maison a été brûlée (The house was burned).