Learn Finnish: Me rakentaisimme oman talon. - We would build our own house.

The verb is rakentaa (to build). First, find the verb stem by dropping the final letter: rakenta-. Next, add the conditional marker -isi-. When the vowel a meets the i from the marker, the a drops out, giving us the conditional stem rakentaisi-. Finally, add the personal ending for "we", which is -mme, to get rakentaisimme (we would build).

Why do oma and talo end in -n here?

oman talon is the object of the sentence (the thing being built). In Finnish, when you are doing an action to a whole object and completing it—like building an entire house—the object takes the -n ending. This is known as the total object.

The English translation says "our own house", but where is the word for "our"?

The word oma (own) gets its context from the verb. Since the verb is rakentaisimme (we would build), it is automatically understood that it is our own house. You could add a possessive suffix to the noun to make it oman talomme, but just saying oman talon is perfectly clear and very natural in Finnish.

Is the word Me strictly necessary in this sentence?