A 3D mesh is the structural build of a 3D model consisting of polygons. 3D meshes use reference points in X, Y and Z axes to define shapes with height, width and depth.
While it can take large numbers of polygons to make a 3D mesh approach photorealism, these relatively simple shapes allow for faster processing than other techniques, like NURBS, that produce smooth curves. The polygons used are typically quadrangles or triangles; these geometric shapes can be further broken down into vertices in X, Y, Z coordinates and lines.
Most 3D meshes are created by artists using software packages -- commercial suites like Maya, 3D Studio Max or the free open source Blender 3D. When models are created for animation, they require especially careful construction; odd deformations can result unless polygons are carefully laid for continuous edge loops (a system connecting all edges) around areas that will be moved.
3D modeling can sometimes be a smooth process like working with clay. However, the necessity for edge loops often involves methodical and laborious placement of polygons. Some modelling software includes a process for defining the polygon layout separately from making the general shape.
How to create a 3D mesh from a simple graphic in Blender: