A counselor is a member of a presidency in the LDS Church. Usually there are two counselors in a presidency: First counselor and second counselor, both acting under the direction of a president. Presidents typically delegate tasks or roles to their counselors. For example, a bishop (not considered a president, but fitting the same general role) might ask one of his counselors to advise his ward's sunday school program, and the other to look after the ward's missionary efforts.
When a president cannot attend an important meeting or function, he or she will typically ask a counselor to attend.
Both men and women fill "president" roles in the LDS Church. An example of a female president is a Relief Society president.