On the Solemnity of Pentecost, The Church and Christians commemorate the descent of the Holy Spirit upon the Apostles, gathered around the Mother of the Lord.   The most significant manifestation of the Holy Spirit in Sacred Scripture is recorded at the beginning of the Acts of the Apostles, chapter 2. 

The first Pentecost is seen as the day the Church, newly endowed with the fullness of the Holy Spirit, began her sacred mission of evangelization mandated by Christ.

The Church's liturgy celebrates the Solemnity of Pentecost fifty days after Easter Sunday of the Resurrection of the Lord, Jesus Christ.  Pentecost (from the Greek pentecoste meaning “fiftieth”).

What did Jesus promise His Apostles before He ascended into Heaven?

After Jesus' Resurrection and before he ascended into heaven, Jesus ordered the Apostles “not to depart from Jerusalem, but to wait for the promise of the Father” (Acts of the Apostles 1: 4-5), then he asked them to stay together to prepare themselves to receive the gifts of the Holy Spirit. 

The "Promise of the Father" is the Holy Spirit.