Let's say I connect to a remote computer with TeamViewer. On the remote computer I open an application that requires a password to run. The password field is masked, i.e. as I type it displays ******. Is it possible for the remote user to somehow log or otherwise obtain the incoming keystrokes and therefore the password?
Can the remote user log the incoming keystrokes and therefore the password?
Yes, if there is monitoring software (for example a keystroke logger) enabled on the remote machine.
Teamviewer indicates they secure their connections client to client.
If implemented properly this should mean no one can intercept the data you send over the network in plain text, i.e. find you password and username, because the connection is encrypted.
In other words, in the case you describe (you connecting to your home computer) it should be secure, assuming you don't have any malware/keylogger running on your pc (but again: that has nothing to do with Teamviewer).
The same may not be true if you connect via Teamviewer to a client his computer. Not because the connection is not secure, but because the PC you are connecting to might have monitoring software enabled.