What does Matthew 16:24 mean?

Jesus has said that He must be killed by the Jewish religious leaders before being raised on the third day (Matthew 16:21). Now He introduces the idea of crucifixion, though He does not speak of His own. That's something Jesus will allude to, regarding Himself, later in His ministry (John 12:32–34) when He speaks of execution on a cross in relation to those who would follow Him.

The Roman empire used the horror of crucifixion as one way to keep conquered populations in line, executing criminals and rebels from the lower and slave classes by the thousands. Modern culture often forgets that crucifixion was about much, much more than physically pain. It was an act of violation, public humiliation, control, and torture. Victims were typically stripped naked and beaten. The condemned were forced to carry the crossbeam themselves to the waiting upright part of the cross from which they would be hung. Mockery and insults were flung at them along the way. The crucifixion process was carefully arranged to maximize suffering and draw out death, sometimes taking days. The dead were often left to rot in public view.

In short, references to crucifixion in the ancient world evoked feelings of humiliation, dishonor, misery, and shame. Every person hearing Jesus' words likely had a vivid picture of this in their minds. That would have made it jarring to hear Jesus use crucifixion in comparison to following Him.

Self-denial was a common enough idea in both Jewish and Greek teaching of the day. Jesus, though, pictured a self-denial that included willing participation in the death of self. In short, nobody who follows Jesus can hold on to even the smallest bit of their own agenda, their own dreams, their own way of living in the world. They must sacrifice every ounce of self if they would choose to walk after Him.