The default of vsftp is actually just the opposite, it doesn't chroot users unless you change the default config. Sort of odd for the security king FTP server, but that is the way it works. Of course given the permissions system, it does make some sense.
To the OP, you won't be able to upload files to /var/www until you make the changes the previous responders said. Typical users do not have write permissions for /var/www, so you will need to work with the configs of apache and vsftpd so that you have one user or group with read and write permissions for the directory where the website exists. Usually /var/www is either root:root, or www-data:www-data, which is the default owner of apache2 on most systems.
You should search here for how to chroot users in vsftpd. It is asked and answered all the time. Set the owner of the directory where the website is, make sure the permissions are square, and then also set that as their home directory. Then enable the chroot, and the site user (do NOT use your account, or you will be very sorry) should be able to upload/download files to and from the website. Also, if the site directory is owned by root:root, you'll have to change that, as root can't ftp through vsftp, because that is a major security risk.