Windows by default will allow all administrators (domain admins, local admins, etc...), but no one else. To allow non admins you need to add them to the "Remote Desktop Users" group. You can do this easily enough via control panel (if you can remote in as an admin on the machine), but if you have a hundred machines, the following becomes more friendly (and much quicker, if you use the up arrow to retrieve the last command in your command line history). You will need psexec.
psexec -h \\remote-pc -u domain\administrator net localgroup "remote desktop users" domain\user1 /add