The Republican (Dougherty) is serving an extended term right now. Her term expired several months ago. Obama nominated Thomas Beck (another republican) to the seat but the Senate is sitting on the nomination. Dougherty continues until a successor is confirmed. By law, the Board has to have no more than 2 members of either party. That's why Obama had to nominate a Republican.

Puchala's term expires summer 2012 and Hoglander just got re-appointed for his 3rd term (I think he is good til 2014).

Puchala and Hoglander are the ones who are driving the bus right now. They are the ones who changed the voting rules that upset Delta management (Dougherty dissented) and the ones who are keeping these "interference" claims alive and blocking the flight attendant integration.
Puchala is a former chairman of the flt attendant union (AFA) and Hoglander is former TWA MEC Chairman.

The terms are for 3 years.

Management can't "walk away" for months unless the NMB allows it. Either side can request NMB mediation and then the negotiations proceed on a schedule set by the mediator. However, the mediator in the APA talks has essentially allowed AA management to walk away. The talks were "parked". (to use the colloquial term). It does happen. That's why APA called in ALPA. You know the APA had to be pretty screwed up if even a Democrat majority NMB went along with parking them.