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I would like to know what the difference is between the Topology Checker and Check Validity tools?
Is there anyone who knows of a good document or tutorial that explains functions of these tools? One that, if possible, has figures of vector topological errors.
Topology means "the rules and behaviors that model how points, lines, and polygons share coincident geometry." (Source.) These rules can apply specifically to geometry and how it is stored, or be created by you to check for certain things.
Topology Checker looks at rules you have set up or chosen to apply to the data. Things like a line must be inside a polygon on a particular layer, or there should be no gaps between polygons on one layer as compared to a boundary on another. The link there describes rules available in the QGIS Topology Checker plugin, but doesn't have graphic examples. Esri produced a handy pdf chart with graphic examples that describes their available topology checking rules. Many of the rules are available in both, but regardless it can be helpful to understand the different rules that can be considered.
Check validity looks for geometry errors. You mention GRASS, which has some specific topology rules based on how it stores geometry. Geometry errors include things like self-intersecting polygons (bowties for example), correct order of vertices (ring direction for a polygon), or duplicate vertices. Again I can't find much in the way of QGIS specific graphic examples, but Esri has a few pages about geometry errors, including this blog post. It's worth noting that Arc and QGIS checkers don't always agree on what is or isn't a geometry error (and not just because of using the GRASS format - shapefiles can pass one and fail the other). I found one graphic example in QGIS at this blog post.
Hopefully this answer and those links will explain what you need, but please comment if I can (try) to clarify anything specific.
