To say it in very simple terms, SVN allows you to store your files on a server with a change history and have a "working copy" on any computer you like. You only work in your working copy and at some intervals tell SVN to copy the changes you make to the server. SVN will then overwrite the files on the server, but at the same time keep a record of what has changed. This means, that you can always go back to some earlier version – no more need for manual backup!
Also, SVN is great for working in groups. Because the files are on the server and everybody can have his own working copy on his own computer, you need not send around files with the changes you make. Every group member just makes her changes whenever she is ready to get the changes to the group, she just tells SVN to copy them to the server. The other group members only have to update their working copy with the newest version on the server and all have the same version of the code.
That’s actually about it, if you only want to use the basic functionalities. Just some terminology: Creating a working copy is called "checkout", copying code from the server to your working copy is called "update" and copying your changes from your working copy to the server is called "commit".