Sorry for the two month delay. Anyways, let's get down to restoring a backup. Let's assume the scenario is in a new host, as this was intended to restore at a place where there's nothing. Let's get to restoring the database first.
Restoring the Database:
1.) Create a new database on your new host. You should know how to do that.
2.) Go to phpMyAdmin and click on your database on the left.
3.) Click on Import on the next page.
4.) Click on browse and find your database (circled in red). Hit go. You should be done after that if it said Import succeeded. Bigger databases require re-uploading due to timeout reasons.
Restoring the Files (Method 1):
1.) Go to File Manager. Then go to the directory you will upload your forum at and upload.
2.) Hit browse, and then upload your files. NOTE: Check the maximum upload limit. If your zipped file (zip, tar.gz, etc) is GREATER than that, stop and use Method 2.
3.) Click on the zipped file and then hit extract.
4.) Delete the zipped file. You're done! Move on to CHMOD/Reconfigruation.
Restoring the Files (Method 2):
The second method uses FTP. I won't go into how to use FTP, since most people should know how to use it. If not, there are many tutorials on the internet. Simply extract your files and upload one by one.
The last step is reconfiguration. This is the trickiest part. You'll have to reconfigure the settings of your configuration file. Here are a few depending on software. You have to change the database name, username, and password.
For phpBB - Edit the config.php file found in the forum root.
For SMF, edit Settings.php in the forum root.
For MyBB, edit config.php located in the inc folder.
For IPB 2.3/3.0, edit conf_global.php
For vBulletin, edit config.php located in the includes folder.
You will also have to re-CHMOD all the required files and folders. You should consult with your forum software for a list of that.
Anyways, you're done! Hope this helps.