Are Jews a race?
No, Jews aren't a race - they're a religion (followers of Judaism are called Jews).
No. Judaism is typically considered to be an ethno-religious group. It has certain elements that are common to all ethnicities, such as a common language, a common ancestry, particular customs of association, and a view of common historic nation-hood. It also has certain elements that other religions share such as a belief in divine beings and a God, specific divine mandates, houses of worship, and holy scriptures. (Other ethno-religious groups include the Druze and Yazidi.)
Some people like to say that Judaism is similar to a nationality since "nationality" bears many of the traits of an ethno-religious group. Ethno-religious groups, unlike other common religions, prevent a person from converting out. This is because the ethnic component of Judaism ties a person to Judaism regardless of what they believe. This is as opposed to a pure religion, like Christianity, where failure to believe in the Christ makes a person a non-Christian. However, a person can gain a nationality through a difficult process of citizenship-acquisition. This is because the religious component of Judaism accepts conversion. This is as opposed to a pure ethnicity, like Italian. A person cannot convert to Italian as much as he may like spaghetti and pizza.
If you are referring to race as a unique genetic group, then no. There are Jews of every race and when a person converts to Judaism following the laws of conversion, they are considered as if they were born a Jew. Additionally, according to Jewish law (this is not accepted by the Reform movement), if a child is born to a Jewish man and a non-Jewish woman, they are not a Jew in any sense.