Vertices or "verts" are the basic building blocks of meshes in blender. They are zero-dimensional points that, when connected to other verts by edges and faces, define the shape and size of a mesh. They are also represent where a halo particle is located, if the mesh has the Halo setting turned on. Lone or unconnected verts are not visible when rendered unless they are using the Halo setting.
Vertices have no size. They are simply locations in the 3D "space" inside the Blender file. They can only be seen in Edit Mode, where they appear as pink dots (if not selected) or yellow dots (if selected). However, when multiple verts are combined to form a mesh, the shapes they define can be seen in other modes and can be rendered.
There is, however, one way a vert can be visible when rendered or when viewed in a mode other than Edit Mode. If the mesh has the Halo setting turned on, edges and faces become invisible and verts are rendered as glowing particles.
Vertices are used to determine the shape and size of a mesh. Simply moving a single vert can dramatically change the shape of the object it's a part of. Groups of verts can also be moved, scaled, or rotated to change a mesh's appearance in many ways. Individual verts, however, can only be moved, because, as stated earlier, they are simply points in space and have no shape, size, etc. Only multiple verts selected at once can be scaled and rotated because, unlike individual verts, a group of verts does have a size and shape rather than just a location.
Verts are also necessary for making edges and faces, the other two building blocks of 3D meshes. A pair of verts can be connected to make an edge, and 3 or 4 verts can be connected to make a face.