What is a Hebrew name?
If you Google it, you may be directed to a Wikipedia page discussing names of Hebrew origin. This is exactly on the wrong track. While Hebrew names may be 'of Hebrew origin' (or Yiddish or Ladino or Hebreo-Arabic), they are not just any name.
A Hebrew name in a Jewish context refers to the ritual name given at a bris (religious circumcision) or baby naming to a Jewish child or to a person becoming Jewish upon completing the beit din. It is used during religious rituals, like being called up to bless the Torah, getting married, or other formal ritual events.
For American Jews, it is commonly a different name than their given names. For instance, a Jew with a secular name that is something like Peter Edward Goldfarb might have a Hebrew name that is Pinchas Elifaz. The first letters often correspond. Instead of listing the Family name, it is traditional to list the parentage - a Hebrew name is not complete without this component.
Traditionally, a Hebrew name would follow the formula Hebrew Name ben (son of) or bat (daughter of) Father's Hebrew Name. As Jewish communities have become more aware of the need to recognize mothers as well as fathers in parentage, the mother's Hebrew name has been added. For most progressive Jews this is customary but not required.
For Jews from Israel and much of the rest of the world, it is more common to be given a name that is both one's secular name and ritual name.
How do I find out what my Hebrew name is?
The best source would be a bris or naming certificate or a bar or bat mitzvah certificate. It should have your name with parentage written in Hebrew. This is especially helpful because there are many slightly different names that sound close, but have different meanings. If you don't have either of these, ask your parents. They may have a story that clarifies the meaning, making the appropriate spelling obvious.
I'm Jewish but I don't have a Hebrew name?
Consult your rabbi. They will have the most resources to help you decide the best course. They may help you choose new Hebrew names. For your ketubah, you might want to choose writing your secular name phonetically in Hebrew (transliterating) especially if you are marrying a non-Jew.
I'm not Jewish and I'm not converting at this time
For your ketubah, you might want to choose writing your secular name phonetically in Hebrew (transliterating). This is commonly what people do.