EOD officers and enlisted personnel train together in the basic EOD pipeline, which takes about one and a half years. New EOD officers typically go to a major Fleet homeport (San Diego, Norfolk, Pearl Harbor, Jacksonville, etc.) to a mobile unit. There are a wide variety of assignments from their mobile unit, going in small detachments on short-term missions, deploying with another part of the Navy for an extended period - lots of different things. Warfare officers, in general, tend to follow a rotation of operational/sea duty then rotate to shore duty, gathering up more professional qualifications, education, leadership and staff officer experience along the way, all leading to opportunities to command as a more senior officer.
As it is with all officers, an EOD officer is responsible for executing the assigned mission by directing his or her enlisted EOD technicians in the performance of their duties, managing resources, making decisions, setting priorities, and so on.
The official Navy.com website, unofficial wikepedia and unofficial ehow sites all have more information.