The mesh is a collection of elements which serve to break-up the CAD model into smaller structural pieces - each piece (or element) can be thought of as a little spring with a three-dimensional spring equation, and the collection of elements represents a structure. The 'solver' will determine the total elastic behavior of the structure when you hit the "run" button by solving the spring equation in each element, and collectively for the whole structure. Generally speaking, the more elements in the structure, the better it will approximate "real life" behavior of the whole structure.
A little more detail:
For structural mechanics, the elastic behavior is "modeled" via mathematics using the mesh, which is defined by the "elements" that are located throughout the volume of the CAD model. (very) simply put, each element's structural behavior is modeled as a three-dimensional spring with a force balance equation applied to each element, such as F = KX. This looks similar to the spring equation you learned in physics courses. Here, "F" are the external forces on the element, K is a spring stiffness for the element, and X are the displacements at points around the element. The capital letters are symbols representing a collection of forces, spring stiffnesses, and deflections at different spatial points in and around each element (since they are "3D").
Since elements are generally connected to other elements (at points called "nodes" which are not shown in SW), the forces you apply to a CAD model surface can get transmitted throughout the whole CAD model and reacted at a "fixed" boundary condition that you might apply on another CAD surface, for example.
There's a lot more to this in terms of practical applications, "meshing" technology and tips, stability, and so on, but you can find a lot of material on the internet. A focused course is best. I have seen FEA courses at NPTEL (National Program on Technology Enhanced Learning) for FREE: http://nptel.iitm.ac.in/.
The NPTEL programs are excellent, esp. if you can focus and dedicate time to the course. I have sat through a few of these in other subject and I am very, very impressed.
Hope that helps a little.
Wow, that was very thorough and informative. Thank you so much. And thank you also for putting your answer in layman's terms