What is the meaning of the ${0##...} syntax with variable, braces and hash character in bash?

See the section on Substring removal in the Advanced Bash-Scripting Guide‡:

${string#substring}

Deletes shortest match of substring from front of $string.

${string##substring}

Deletes longest match of substring from front of $string.

The substring may include a wildcard *, matching everything. The expression ${0##/*} prints the value of $0 unless it starts with a forward slash, in which case it prints nothing.

_{‡ The guide, as of 3/7/2019, mistakenly claims that the match is of $substring, as if substring was the name of a variable. It's not: substring is just a pattern.}

Linux tip: Bash parameters and parameter expansions

${PARAMETER##WORD}  Results in removal of the longest matching pattern from the beginning rather than the shortest.
for example
[ian@pinguino ~]$ x="a1 b1 c2 d2"
[ian@pinguino ~]$ echo ${x#*1}
b1 c2 d2
[ian@pinguino ~]$ echo ${x##*1}
c2 d2
[ian@pinguino ~]$ echo ${x%1*}
a1 b
[ian@pinguino ~]$ echo ${x%%1*}
a
[ian@pinguino ~]$ echo ${x/1/3}
a3 b1 c2 d2
[ian@pinguino ~]$ echo ${x//1/3}
a3 b3 c2 d2
[ian@pinguino ~]$ echo ${x//?1/z3}
z3 z3 c2 d2