The problem is that the backup is trying to access the network folder with the credentials under which SQL Server is running. In other words the backup is made by SQL Server, not by the user who executes the backup command.
You can use (create a new) domain account for SQL Server service and give proper privileges to that account if you want to take the backup directly to network. Another option is that you take it locally and then using some script transfer the backup to network share.
One example article dicussing setting the privileges for the domain account:
http://support.microsoft.com/kb/283811[
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If the thing mentioned above is already correct, check the backup settings and the target folder. For example if the backup tries to write a new file (without overwriting the old) and there's an existing file with the same name in the folder, it would fail.