Hebrew names are used in prayer in and out of synagogue and for other religious rituals. When a person is called up in synagogue for an Aliyah (the honor of reciting a blessing over a Torah reading), he is called up by his Hebrew name. The names that appear on a Ketubah (marriage contract) or on a get (writ of divorce) are Hebrew names. When a people are ill and mi shebeirah prayers are recited for their well-being, they are identified by Hebrew names. When a deceased person is remembered through the Yiskor prayers recited on certain holidays, the Hebrew name is used. Jewish tombstones sometimes carry the Hebrew name instead of or side-by-side with the secular name.
A Hebrew name begins with a given name, followed by ben (son of) or bat (daughter of), followed by the person’s father’s Hebrew name and the person’s mother’s Hebrew name. If the person is a kohein (descendant of Aaron), the name is followed by “ha-Kohein.” If the person is a Levite (descendant of the tribe of Levi), the name is followed by “ha-Levi.” If the person or his father is a rabbi, some follow the name with “ha-Rav.” This format of naming is seen as early as the Torah where, for example Moses’ successor Joshua is repeatedly referred to as Yehoshua ben Nun (Joshua, son of Nun). Note that the surname is not the same from generation to generation: Abraham’s son Isaac is Yitzchak ben Avraham; Isaac’s son Jacob is Ya’akov ben Yitzchak, and so forth. Moses’ Hebrew name would be Moshe ben Amram ha-Levi (because he is a member of the tribe of Levi but not a descendant of Aaron), while his brother Aaron would be Aharon ben Amram ha-Kohein (because Aaron was a priest).
The secular name usually corresponds in some way to the Hebrew name. Sometimes, the name is exactly the same or an Anglicized version of the same name: David, Michael or Sarah are as good in Hebrew as they are in English, though they are pronounced differently. A person with the Hebrew name Yosef would probably have the English name Joseph and Rivka might be in English Rebecca. Sometimes, the English name retains only part of the Hebrew name, for example, Aharon might become Aaron in English, but it might also become Harry or Ronald. Sometimes, the English name retains only the first letter of the Hebrew name: Pinchas becomes Philip or Nechama becomes Natalie.
There are no hard-and-fast rules about how to translate Hebrew names into English, and indeed, there is no real reason why a person’s secular name has to correspond to the Hebrew name at all.