Binder refers to substances that hold the particles of pigment together in paint. All paints include a binder of some sort because this is what keeps the pigment in place after the paint dries.
Acrylic painting mediums generally include a synthetic binder designed to form a film after water has evaporated. Meanwhile, oil painting workshops will tell you that oils use organic binders, traditionally linseed oil. As for watercolor painting, that requires paints that use gum arabic as a binder.
Binder vs. Vehicle
Sometimes the term is used interchangeably with vehicle, although this is technically incorrect because it is merely part of the vehicle component of a paint. In oil paints, the binder constitutes the whole vehicle, whereas acrylics may have up to 10 components of the vehicle, of which the binder is only one.
See also: Egg Tempera