Learn Spanish (Spain): ¿Dónde te alojaste en Madrid el mes pasado?

The verb used here is alojarse (to stay or to lodge). It is a reflexive verb, which means it needs a reflexive pronoun. Since the sentence is asking what you (informal) did, we use the pronoun te along with the 'you' form of the verb in the past tense, alojaste.

Could I use te quedaste instead of te alojaste here?

Yes! Quedarse is another reflexive verb that means to stay. Saying ¿Dónde te quedaste...? is very common and sounds a bit more casual. Alojarse specifically refers to lodging or staying in an accommodation like a hotel or apartment.

Why does it say el mes pasado instead of just mes pasado for 'last month'?

In Spanish, you generally need to include the definite article (el or la) when talking about specific past or future times, like el mes pasado (last month) or la semana que viene (next week). English skips the 'the', but Spanish requires it here.

How do we know alojaste is in the past tense?

It uses the preterite tense, which Spanish uses for completed actions in the past. The root verb is alojar (an -ar verb), and the -aste ending specifically tells us it is the past tense for tú (informal you).