Learn Portuguese (Portugal): Ele está apaixonado pela vizinha.

In English, you are "in love with" someone, but in Portuguese, the expression is estar apaixonado por someone. Here, por combines with the feminine "the" (a) to become pela, because vizinha is a feminine noun.

Why is it está (from estar) rather than é (from ser)?

The verb estar is used for conditions and states, including emotional states like being in love (apaixonado). Even if he has loved her for a long time, it is still treated grammatically as a state he is currently in.

The English translation says "his neighbour", but there is no word for "his" in the Portuguese sentence. Why?

In Portuguese, it is very common to drop possessive words like sua (his) when the context makes it obvious. Saying pela vizinha (literally "by the neighbour") is the most natural way to say "his neighbour" in this context.

If the sentence was about a woman being in love, what would change?

The adjective describing the feeling must match the gender of the person experiencing it. If it were a woman, the sentence would be Ela está apaixonada pela vizinha.