How can I chmod 777 all subfolders of /var/www?

This is bad practice, but hopefully you are just using this for development, or you have another good reason. You can specify the permissions when you create a directory using the -m option:

mkdir -m 777 dirname

Or you can set the permissions recursively.

sudo chmod -R 777 /var/www

Before using either of these, really consider if you want your filesystem to be so accessible.

Edit: As mentioned by Rinzwind here is a better way of accomplishing what you want.

Check what group owns your /var/www directory and add your user to that group.

sudo adduser yourusername group

The group is probably www-data.

Then you will be OK with setting your permissions to 775.

Files and directories in Unix may have three types of permissions: read (r), write (w), and execute (x). Each permission may be on or off for each of three categories of users: the file or directory owner; other people in the same group as the owner; and all others. To change the mode of a file, use the chmod command. The general form is chmod X@Y file1 file2 ...

chmod a-w file (removes all writing permissions)
chmod o+x file (sets execute permissions for other (public permissions))
chmod u=rx file        (Give the owner rx permissions, not w)
chmod go-rwx file      (Deny rwx permission for group, others)
chmod g+w file         (Give write permission to the group)
chmod a+x file1 file2  (Give execute permission to everybody)
chmod g+rx,o+x file    (OK to combine like this with a comma)

u = user that owns the file
g = group that owns the file
o = other (everyone else)
a = all (everybody)

r = read aces to the file
w = write access
x = execute (run) access