In addition to the grade, plywood can have different cores.
The following is taken from PaxtonWood.Com
There are two types of Combination Core construction. The first type consists of a center veneer core and a crossband veneer on both sides of the center with a layer of MDF, particleboard, or hardboard under the face and back veneer. The second type consists of an oriented strand board or wafer board center with a veneer crossband on both sides, under the face and back veneer. Combination Core can be used when the project calls for consistent flatness and good screw-hold ability, without the weight of a MDF or particleboard core.
Lumber edge glued into a solid slab is considered Lumber Core. Lumber Core plywood is most often found in the construction of import panels, usually only 18mm thickness panels. Lumber Core plywood consists of a face & back veneer, then a layer of crossband veneer, with the edge glued lumber in the center. Lumber Core plywood should be used for the bending strength and screw-hold ability. The ability of Lumber Core plywood to bounce back makes it an excellent choice when constructing long or wide shelves in cabinets or closets.
Medium Density Fiberboard Core has the most uniform thickness and consistency of any panel core. The face and back veneer are glued directly to the MDF Core without the need for a crossband veneer. The consistency in thickness and flatness makes MDF Core panels the preferred choice for 32mm construction.
Particleboard Core panels are constructed similar to MDF Core panels, with the face & back veneers glued directly to the core. Particleboard Core lays flat like MDF, but does not have the screw-hold strength of other cores. Particleboard Core is the least expensive of all core types.
Veneer Core plywood consists of a center veneer and crossband veneers alternately layered on both sides of the center to the desired thickness producing an odd number of veneers for the core. The weight-bearing strength of Veneer Core plywood is excellent along with its bending strength and screw-hold ability. Veneer core panels are lighter in weight than any other core type.
In addition, the veneer on the panel can be cut one of two ways.
Plain slicing a log occurs when a log is only turned a few inches at a time when slicing it, versus a constant turn and peeling method used for Rotary Cut veneer. Plain Sliced veneer produces a cathedral grain pattern, similar to the pattern produced when plain sawn lumber is produced. Plain Sliced veneer is cut along the growth rings, producing the highest yield of any “slicing” method.
Rotary is the process of peeling the entire long. Turning the log, while shaving the veneer as it turns produces Rotary Cut veneer. This process is the only cutting method that is capable of producing whole piece face veneers. Rotary cut will yield veneer with a broad grain pattern with no plain sliced or quartered appearance. Rotary cut veneers are used in the majority of panels produced in North America. This process yields the most veneer per log and is usually less expensive than sliced veneer.