Each vertex in a mesh can be assigned to one or more vertex groups. In each group it has a value between 0.0 and 1.0 called a vertex weight. Use this tag for questions about the use of vertex weights.

The manual says this about Vertex Weights:

each entry in a vertex group also contains a weight value in the range of (0.0 to 1.0).

That is, the vertices in a mesh can be collected into groups of vertices called Vertex Groups and within each Vertex Group they can be assigned a weight. The main use of Vertex weights is to adjust the deformation of the mesh in response to deformation events such as armature movement.