In an article on MyJewishLearning.com, Rabbi Louis Jacobs writes that many Jewish people do celebrate birthdays. Although this practice is adopted from the secular world, Rabbi Jacobs notes that questions of Jewish doctrine are not involved with birthday celebrations.
Many Jewish people, especially Orthodox Jews, do not celebrate their birthdays, according to an article by Ted Merwin in The Jewish Week. He explains that birthday celebrations were not a Jewish custom, other than ceremonies marking rites of passage, such as the third birthday when a male child received his very first haircut, the fifth birthday when he would begin to study the Torah and the bar mitzvah at the age of thirteen.
Merwin goes on to explain that some ultra-Orthodox Jews are taking a more positive view of birthday celebrations, and notes that some rabbis recommend reciting the Shecheyanu before cutting a birthday cake. The Shecheyanu is a prayer that thanks the Lord for allowing a person to reach a milestone.