Unbounded (complete) minimal surfaces are never compact. That means one can never make a sphere large enough such that no part of the surface falls outside of it *. This property may take two forms: periodicity or ends. (Note that triply periodic surfaces are sometimes considered compact in the quotient.) Despite the existence of numerous minimal surfaces with ends, the ends fall into a few types.
* To see why this is, imagine a compact minimal surface. Take an enclosing sphere and shrink it until it touches the surface at some point. Then the surface must have a tangent plane at that point. But of minimal surfaces, only the plane has a tangent plane.