I copied your script, just to make sure I was reading it right. Here's what I found...
read -p "What month is it?" month
$month <<<<<<<< Remove me
case $month in
"Janurary") echo " $month is Winter ";;
"Feburary") echo "$month is Winter ";;
"March") echo " $month is Spring ";;
"April") echo " $month is Spring ";;
"May") echo " $month is Spring ";;
"June") echo " $month is Summer ";;
"July") echo " $month is Summer ";;
"August") echo " $month is Summer ";;
"September") echo " $month is Fall ";;
"October") echo " $month is Fall ";;
"November") echo " $month is Fall ";;
"December") echo " $month is Winter";;
*) echo " $month is not valid" ;;
esac