I am somewhat struggling to see the difference between a regular CW complex and a non-regular CW complex.
The difference is all the attaching maps are homeomorphisms - i.e. there are no identifications made on the boundary. So I guess if I produce a 1-sphere (circle) by a single zero cell and a single one cell, this is not regular (as both endpoints of the 1-cell get mapped to the zero cell)? However, if we use two 1-cells and two 0-cells we can get a regular CW structure?
I guess this is not regular (the 2-cell intersecting the 1-cell at the top is the problem.
The 'thoughtful' question coming from this - we have seen the sphere admits both a regular and non-regular CW complex. To me, the regular CW complex seems easier to work with, as the "degree term" in the cellular boundary formula is either $-1,0,1$.
What type of spaces admit a CW structure, but not a regular one? I am thinking of a pathological example, such as attaching a 2-cell to the 1-sphere with attaching map like $x \sin(1/x)$ (what would that look like?!)