Well, first, to be pedantic, "crest" refers to the Mermaid that sits on the helm ("helmet").
A coat of arms is awarded to an individual and may be passed to down through his male descendants - they do not belong to just anyone who happen to have that surname.
In theory, no one else should have those arms.
The coat of arms as pictured here I'm guessing was simply lifted out of Burke's by the person who made it - a very common commercial practice. This may mean a number of people with the same surname apparently have the same arms even if the arms have never been granted to their specific family.
I'm sure there are examples, but it's rarer still for two people with different, unrelated, surnames to have the same arms, as even people selling commercial products with "your" coat of arms avoid this, while the College of Arms that designs the "official" arms for an individual exists in part to avoid just this sort of thing (in theory).
If your family was indeed granted a coat of arms, you'd probably have known about it.
According to Burke's (1884), these arms were granted to the O'Byrnes family from Co. Wicklow in Ireland - similar arms were apparently still in use in 1884.
Of course, I'd be delighted for you should you tell me that you've traced your ancestry to the O'Byrnes of Co. Wicklow!
