Pictorally, you can think of a loop in a space as being like noose wrapped around part of the space; if the space is simply connected, you can always be sure that you can pull the noose down to nothing.

For instance, the surface of a doughnut (called a torus) is not simply connected, while the surface of a sphere is.

An equivalent formulation is that a space is simply connected if and only if its fundamental group is trivial.