The notion of operad comes in two broad flavors (apart from the choice of enriching category): planar operads and symmetric operads. Accordingly, planar operads are also called non-symmetric operads. Another term is nonpermutative operads.
A planar operad is a collection (set/object in some enriching category) of -ary operations for all , equipped with a suitable notion of composition. In contrast, a symmetric operad in addition carries an action of the symmetric group on the object on -ary operations, and all structures are required to respect this action.
The notion of planar operads takes its name from the fact that the operations in a planar operad may naturally be drawn as planar trees without, in general, a relation between two trees that cannot be related by a planar deformation into each other.