Since I don't often get to see tugboats, it is pretty cool when I do. It's kind of amazing that a small vessel like a tugboat is named after a large mythological being like a Titan. Sadly, I don't have many details on the tugboat nor where she was heading.
In Greek mythology, the Titans were the second generation of deities. They proceeded the Greek gods that everyone knows about (Zeus, Poseidon, Athena, etc) and succeeded the primordial gods (Gaia and Uranus).
There were 12 Titans and they ruled with great strength. The more famous Greek gods were the children of Cronus and Rhea. Like the Titans overthrow their parents, they were in turn overthrown by the Olympian gods.
Because we owe so much to Greek Mythology, there are several things called Titan.
There is the moon of Saturn named Titan. It is believed to be the thing closest related to the Earth in the solar system. There are even theories that it has the building blocks of life. Although, I'd have to believe that if there were life, it wouldn't look anything like life here.
There is also the Titan missile. The Titan missile was one of the earlier US intercontinental ballistic missiles. I believe it was capable of carrying a 1 megaton warhead. It was also used in our early space program.