You get dropped off in front of the terminal, where you hand over your checked luggage to the porters. You then enter the terminal and go through a security screening with your carry-on bags: x-ray machine and metal detector walk-through. You then go to the appropriate check-in line, which is usually based on deck number, or if you are in a suite. They will have signs. These are cattle chute style with the maze-like layout of rope barriers to control crowding. Whether your line is long or short really depends on when you get to the pier. Early means more crowding and waiting. Later can often mean no lines and you go from curb to the ship almost non-stop.
Once you have checked in with one of the desk agents, if the ship is open for boarding you will proceed directly aboard, with perhaps a stop along the way for a security photo if they don't take it at the check-in desk. If the ship is still not open, you will be directed to a seating area to wait for boarding to commence. When boarding starts they do it by groups to avoid crowding. Thus, you may have to stand at security and then check-in for maybe 10-15 minutes in a slow but usually constantly moving line. Then another 10 minutes to board. It could be a little longer or quite fast, again depending on your arrival time.