What is /var/www/html?

/var/www/html is just the default root folder of the web server. You can change that to be whatever folder you want by editing your apache.conf file (usually located in /etc/apache/conf) and changing the DocumentRoot attribute (see http://httpd.apache.org/docs/current/mod/core.html#documentroot for info on that)

Many hosts don't let you change these things yourself, so your mileage may vary. Some let you change them, but only with the built in admin tools (cPanel, for example) instead of via a command line or editing the raw config files.

In the most shared hosts you can't set it.

On a VPS or dedicated server, you can set it, but everything has its price.

On shared hosts, in general you receive a Linux account, something such as /home/(your username)/, and the equivalent of /var/www/html turns to /home/(your username)/public_html/ (or something similar, such as /home/(your username)/www)

If you're accessing your account via FTP, you automatically has accessing the your */home/(your username)/ folder, just find the www or public_html and put your site in it.

If you're using absolute path in the code, bad news, you need to refactor it to use relative paths in the code, at least in a shared host.