What does Matthew 27:32 mean?

In most cases, Roman crucifixion involved a vertical post permanently attached to the ground. This would make executions much easier and more efficient. Condemned prisoners would typically carry the horizontal crossbeam, which could be nailed to their arms and then attached to the post. Even if the beam was relatively small, it would have weighed around 35 pound, or 16 kilograms. Though Jesus was weakened by that point (John 19:1–4), He apparently managed to carry the beam from Pilate's headquarters (Matthew 27:2) to the city gates. He could not carry it any further, though. The fact that He remained upright at all after the flogging and beatings (Matthew 27:27–31) is remarkable.

The soldiers picked a man coming into the city from the country to carry Jesus' cross for Him. Under Roman law, soldiers could force any subjugated person to carry things for them, for up to a mile. This is part of the background to Jesus' teaching on submission and service for others (Matthew 5:41). Cyrene was a city in North Africa. For this reason, many suppose Simon was of African descent. Cyrene had a large Jewish population, and Simon was very likely in Jerusalem for the Passover, along with so many others. Mark adds that Simon was the father to Alexander and Rufus (Mark 15:21). They may or may not be the Alexander and Rufus mentioned by Paul in Acts 19:33 and Romans 16:13.