For fast usage, you may use:
mv * mywebsite
This will cause an error saying it cannot move mywebsite inside mywebsite, but will move all other files, except hidden files.
If you're using bash you can fix it by enable the extglob via:
shopt -s extglob
then you can use:
mv !(mywebsite) mywebsite
it will move all files except website to website folder.
If you want a more complicated and elegant solution (and also works with hidden files), you may use:
find . -mindepth 1 -maxdepth 1 -not -name mywebsite -exec mv {} mywebsite \;
A last solution can be a loop:
for file in $(ls -1 | grep -v mywebsite); do mv $file mywebsite; done