The default owner of /var/www is root, and you probably don't connect as root.
If you are single user, do
chown -R <username> /var/www
The user will then be owner of /var/www and all it's contents. In this case mode 755 (for directories) and 644 (for files) will do.
If there are more users that should have write access, create a group and assign it to /var/www:
chgrp -R wwwedit /var/www
Add all relevant users to the group, e.g. by editing /etc/group (put a comma separated list of users at the end of the group's line) or use your distro's GUI tool to do so.
In this case, the group must have write access:
chmod -R g+w /var/www
You should also force the group for all new files under /var/www:
chmod 2775 /var/www
chmod 2775 `find -type d`
To avoid permission problems, the users should have a umask of 002 when they come in via console. Default is 022. You can change this (distro-dependant) by editing these user's ~/.bashrc
If you use Samba to access /var/www, set modes 775 and 664 for directories and files respectively for the www-Share in smb.conf.
An alternative is to use the user's "public_html" directory in their home directories, have a look at Apache's httpd.conf.
In this case the web pages can be accessed in the form
Permissions for /home/<user> and /home/<user>/public_html must be set to (at least) 701 for this to work.