A nominee is someone running for office. You could be a nominee for student council or for the presidency.
To nominate is to name or select someone as a candidate for office. In the United States, the primary elections help each party figure out who to nominate to run for office. That candidate is called the nominee. Once a nominee is selected, he or she runs against the sitting president. There are nominees at every level of government. Being a nominee is quite an honor, but it's not the ultimate goal: a nominee could still win or lose the election.