In the first case:
rename ’s/\.bak$//’ *.bak
you are running a regular expression again filenames and replacing matching part of expressions (.bak at the end of a file name) with the second expression (which is empty).
In the second case:
rename ’y/A-Z/a-z/’ *
you are matching against the regular expression pattern space and transliterating to the target. In other words, 'A-Z' is changed to' a-z', making the filenames lower case.
I suggest you look at the man page for sed for more commands and more details. I'll add that I believe the 's' command is used most often. regex (section 7) and perl documentation may also be of help. In particular, here's a tutorial on perl and regular expressions.
Attempt to match regexp against the pattern space. If success‐
ful, replace that portion matched with replacement. The
replacement may contain the special character & to refer to that
portion of the pattern space which matched, and the special
escapes \1 through \9 to refer to the corresponding matching
sub-expressions in the regexp.
Transliterate the characters in the pattern space which appear
in source to the corresponding character in dest.