What happens to us immediately after we die?
The short answer
The body dies but the soul does not. At death the soul separates from the body, is judged at once, and enters heaven (directly or through purgatory) or hell. Each soul is reunited with its risen body at the Last Judgment when Christ returns.
At the moment of death
Bodies die, but souls never do. Every spiritual soul is immortal: it does not perish when separated from the body at death and will be reunited with the body at the final resurrection (CCC 366). So at death the soul leaves the body, is judged immediately, and enters heaven, immediately or through purgatory, or hell. The Catechism calls this the particular judgment, in which each person's life is referred to Christ (CCC 1022). Scripture supports this: Luke 16:22; 23:43; 2 Corinthians 5:8; Philippians 1:23.
At the end
Each soul will be united with its resurrected body just before the Last Judgment, when Christ returns. The Catechism teaches that in Christ's presence the truth of each person's relationship with God will be laid bare, revealing the good each did or failed to do. Only the Father knows its day and hour. Through his Son he will speak the final word on history; we will understand the meaning of creation and salvation and how his Providence led all things to their end. It will show that God's justice triumphs over all injustice and that his love is stronger than death (CCC 1039-1040).