The "~" operator is for matching regular expressions.
$
$ awk '{if ($0 ~ /0$/) {print $0,"Zero"} else {print $0,"number"}}' f1
2323 number
3232 number
4230 Zero
3230 Zero
4340 Zero
4343 number
233 number
32320 Zero
$
$ # or alternatively -
$ awk '/0$/ {print $0,"Zero"} !/0$/ {print $0,"number"}' f1
2323 number
3232 number
4230 Zero
3230 Zero
4340 Zero
4343 number
233 number
32320 Zero
$
$
In addition to the operator error pointed out by durden_tyler, \b in an awk regular expression is an escape sequence which matches the backspace character (unlike in perl where it refers to a word boundary).