Aren't they called mooring lines?
I suppose they are as well, though in the case I was thinking up he wasn't mooring his vessel, he was just using a small tie-up line (it's a tie-up line cause you tie it to things hehe) to pull his bow in closer to the dock in order to paint it. The large 4 inch lines are what comes to mind when I think of a mooring line, in this case it was just a 2 inch line.
I was working on my families boat painting it, and the Time Bandit came and parked behind us, of course a cruise ship happened to come into port at the same time so a deluge of people came down to the docks to go take pictures of it. Our sister ship was across the dock from us, and they were also painting but needed to get the bow. So he hooked a line up to the dock and started winching in on it to pull in the bow.
However, there was a group of tourists standing right next to this line that was getting thousands of pounds of force tensioned on it. They seen the tourists and went to the bow and yelled at them to move away from the line, they acted like they were going to move so they continued pulling on the line. Our sister ship seen they still hadn't moved away, so they went and told the tourists that in that line snapped it would kill one of them, they finally started moving, they got no more than 10 feet away when *SNAP* the line broke and whipped the exact spot they had previously been standing.
They still didn't move very far after this, so I finally said something to them along the lines of what the hell is wrong with you, are you daft or something? They grumbled and walked off.