A cylinder with its side in u0,v0, base in u0,v1, and top in u0,v2
UDIM values are a way of representing the integer coordinates of a texture, from the coordinates of its bottom-left corner in UV space. This way, multiple textures can be used to cover a model, rather than a single, much larger, texture.
Faces of a cube unwrapped in u0,v0, to u0,v5,
It is a 4 digit number starting at 1001, created using the formula, 1001 + u + (10 * v)
In the example above the side would be 1001, base 1002, and top 1003.
To use this method in your file inputs, use the <udim> tag in the file name, which is then replaced with the appropriate number.
texture.<udim>.exr will be read as
texture.1001.exr when looked up for rendering.
texture_<udim>.tx will be read as texture_1001.tx when looked up for rendering.
Check out Ben Neall's page for a clear description of UDIM numbering.