When I connect to my machine and I am on the local network Remote desktop does not require my password. However, when I try to remote into my machine remotely over the Internet, it makes me enter my password each time. Is there a way to have it remember my password, and not prompt me each time?
Password is stored per-address in the RDP settings. Most likely your machine has a different address on the local network and on the internet due to a NAT in between.
What you need to do is to connect your client to the internet. Start up Remote Desktop Connection. Enter the address. Choose Options, then check "Allow me to save Credentials". Then log in as usual and chose to save the credentials.
Note that your password will be saved in clear text on the local system, so don't ever do this on a system that is not fully under your control. There are also a bunch of other security issues with RDP, which is why you should not, if possible, expose it to the internet.
If you really need RDP from the internet a VPN to protect the connection is highly recommended.
Try eliminating any stored passwords for that user and then create a new connection, store that users password once again, an then instead of connection trough a example.rdp file connect trough a shortcut that leads to your server, shortcut leads to: mstsc /V:server