Assuming that you are in the same directory as your files:
for name in awstats.*.conf; do
basename "${name#awstats.}" .conf
done
The code inside the loop will first trim the awstats. prefix off from $name using a standard parameter substitution, and then lets basename trim off the .conf suffix.
You could also do it in two steps without calling basename:
for name in awstats.*.conf; do
newname=${name#awstats.} # trim off prefix
newname=${newname%.conf} # trim off suffix
printf '%s\n' "newname"
done
No files are renamed by these two loops.