There's two answers to the question. As far as the Royal College of Arms is concerned, very few people can lay claim to a coat of arms. Having the same surname as that a coat of arms is related to means zip; you have to be the right sort of direct decendant.
But they can go stuff themselves!
My family name is Williams. There are several coats-of-arms for that surname, and the gap between their being granted and reliable records means it's quite hard to prove one has the right to use it. This is probably NOT my Williams coat-of-arms, it's from the wrong part of Wales for a start and in all liklihood my Williams side of the family has no coat-of-arms.
My paternal grandmother was a Hawkins. It's quite possible but not certain that the family are descended from the Hawkins family of Plymouth, in which case the coat-of-arms is that of Sir John Hawkins;
The lion and the circles ('bezants') are gold, the waves are silver and blue, the backgound is black. The square in the top left is the badge of a Spanish town in Central America that Sir John riaded; interestingly the scallop and palmers staves are similar to that of the Palmer coat of arms Xandria posted; that coat-of-arms is a great example of the play-on-words frequently seen in coats-of-arms; Palmer was an old word for Pilgrim and staves and scallops were symbols associated with pilgrims.
The crest is the shameful bit, as SIr John had the dubious distinction of being the Englishman who started the slave trade; it's a bound slave, and the family have various bits and bobs (a tankard and a ring) with the crest on it.
If we're not decended from that Hawkins (I can get the family tree back to 1700 but then it's guess work), then this is the other Hawkins crest, dating from way earlier;
It's often seen with a normal cross (i.e. not on the diagonal), but this is just because many people drawing the pictures wouldn't know a St. Andrews cross if it bit them.