What does Mark 1:18 mean?

Andrew and Simon Peter's abandonment of their business seems abrupt, but the brothers had already met Jesus. Given Mark's tendency to be brief, it's unlikely the men literally dropped everything and walked away from it, in that instant. More than likely, they made immediate arrangements to leave their business in order to follow Christ. In other words, their "leaving" was initiated "immediately," but the text does not require that it was completed "immediately."

Andrew had been introduced through John the Baptist, and had brought Simon Peter to Jesus in return (John 1:35–42). This slight history, combined with the miraculous haul of fish that Jesus had just arranged (Luke 5:1–11), causes Andrew and Simon Peter to trust Jesus fully. In fact, Simon Peter has such reverence for Jesus that he tells Jesus to leave him because he is unworthy (Luke 5:8). But they both accept Jesus' call. In return, Peter becomes well known as the leader of the disciples and they both remain among the faithful apostles in Acts 1:13 who experience the birth of the church and help lead its early followers.

Mark uses the word "immediately" (Greek eutheos) forty times in his account. Immediately after Jesus came out of the baptismal waters, the heavens open and the Holy Spirit descends (Mark 1:10). Immediately after Jesus spoke, a man is cleansed of leprosy (Mark 1:42). Immediately after Jesus healed a paralytic, he takes his mat and walks out the door (Mark 2:12).

Although Mark shows Jesus acting quickly and His miracles coming to fruition as soon as He speaks, not everything that happens in haste is good. Immediately upon hearing Jesus teach, the Pharisees conspired with the Herodians to destroy Jesus (Mark 3:6). A seed dropped on stony ground sprouts immediately because it has no soil in which to grow roots (Mark 4:5). Acting quickly to Jesus' call is essential, but it's also wise to have a discerning mind.