Plate element versus shell element
Both terms represent the same thing in the STAAD context, which is, a 3-noded (triangular) or a 4-noded (quadrilateral) element to which a thickness has to be assigned as a property. This is commonly referred to as a 2D finite element. In STAAD, this element has both attributes - membrane (in-plane effect) and bending (out-of-plane effect). The bending effect can be shut off by declaring it as ELEMENT PLANE STRESS. The in-plane effect can't be shut off.
There is another type of element in STAAD called a solid element which is a 3d finite element. It is shaped like a prism or wedge or a block or many other shapes that can be formed by 4 thru 8 nodes where no more than 4 nodes lie in one plane.
Plate Element verus a surface
If you want model a structure which contains a wall, slab or panel type component, you have two choices in STAAD :
a) Model that panel using a collection of individual elements. This is called a finite element mesh. This is an assembly of the 2d triangular and/or quadrilateral elements described above.
b) Model that as a single physical object called a Surface.
Option (a) is achieved using the mesh generation facilities in STAAD. Alternatively, you can do the mesh generation using any other software that has meshing capabilities, export the data to a DXF or CIS/2 file, and then import that DXF or CIS/2 into STAAD.
In option (b), (surface object), what happens under the hood is that, during the analysis, STAAD transforms the surface into a finite element mesh. The type of mesh (number of elements, type of elements, size of elements, etc.) that is generated from the surface is based on the parameters that you provide at the time of defining the surface. The details of the mesh thus generated are to a large extent, masked from the user. Results are presented for that surface, not for the individual elements that it is made up of.
To give you an analogy, think of the surface as a chess board, and the plate element as the individual squares in that board. You can define the chess board as 64 elements (option a) or 1 surface (option b).