A billboard is a surface, usually a quad, that is always facing the camera.  

Point (Unconstrained)
A point billboard can rotate whatever way it needs to face the camera.

Axis (Constrained)
A constrained billboard is limited to only rotate around a certain axis.

Geometry Shader Billboards with SlimDX and DirectX 11 - 2013
SpriteBatch billboards in a 3D world - 2011
Volumetric Billboards
Volumetric Billboards
Billboarding and the Geometry Shader