FTP/SFTP not allowing any writing
You should be able to change the permissions (by terminal on the machine or over SSH) by running this so that users other than root can access it:
sudo chmod -R 757 /var/www
or this so that it is owned by your user (which should be defined by $USER):
sudo chown -R $USER:$USER /var/www
You need to do this as only the owner+group can read, write and execute in /var/www, but other users can only read and execute - including the user you are using to try and the folders.
For more info, look at man chmod and man chown