Il trapassato prossimo (the pluperfect or past perfect tense) describes something that had happened or we had done in the past, for example: era partito (he had left), non ti avevo sentito (I hadn’t heard you). Technically speaking, as you can see from the examples, it’s constructed using the auxiliary verbs essere (to be) or avere (to have) which are conjugated in the imperfect tense and followed by the past participle. If you need to revise the use of essere and avere with il participio passato check this link: Transitive and Intransitive verbs

Let’s look at a few more examples in order to clarify its construction:

Siccome non l’avevo vista in giro pensavo che si fosse trasferita (As I hadn’t seen you around I thought that you’d moved away. N.B. you formal)

Mi dispiace, mi ero addormentato/a (I’m sorry, I had fallen asleep)

Alla festa c’era anche il fratello di Lucia. Non l’avevo mai incontrato prima (Lucia’s brother was also at the party. I had never met him before)

Fortunatamente nessuno si era fatto male (Luckily nobody had hurt themselves)

Il passato prossimo (present perfect), on the other hand, is used when talking about the past in general, i.e. about something that happened or that we did, e.g ieri sono andato a Roma (I went to Rome yesterday) or Lunedì è piovuto a catinelle (It rained cats and dogs on Monday). Both the trapassato prossimo and the passato prossimo are often used together in the same sentence. In this type of construction we normally use il passato prossimo or, occasionally, il passato remoto (historical past) to express the most recent event and il trapassato prossimo to express the older event. Let’s see how this works in the following examples:

Avevamo appena finito di cenare quando è arrivato Giorgio (We had just finished dinner when Giorgio arrived)

Mi dispiace, mi ero addormentato/a e non ti ho sentito arrivare (I’m sorry, I had fallen asleep and I didn’t hear you arrive)

Siamo arrivati in ritardo perché il vento  aveva abbattuto degli alberi bloccando la strada (we arrived late because the wind had knocked down some trees, blocking the road)

Eravamo pronti per partire quando Giorgio si è accorto che aveva dimenticato il passaporto a casa (we were ready to go when Giorgio realised he had forgotten his passport at home)

Avevamo deciso di fare una passeggiata, ma poi è cominciato a piovere (We had decided to go for a walk, but then it started to rain)