Sorry if I came off a bit harsh. There is no problem with using PuTTY if it works for you and is what you're used to. If you're interested in learning more terminal stuff, you may be able to do better for your purposes. Infodump follows.
A terminal emulator is the interface through which you access programs like SSH clients and Telnet clients, much like a web browser is an interface through which you access videos, news, etc., so it doesn't make much sense to talk about what it does on its own - it exists to run other programs.
Technically, any program on Linux can access serial devices; even basic commands like cp and cat can talk usefully to serial devices for some simple purposes (e.g. short plotter drawings can just be sent to the serial port with cat, and my plotter will draw them - drawings bigger than the plotter's buffer need something a little more sophisticated).
PuTTY bundles together a terminal emulator, SSH client, and Telnet client. When you connect to something it's running the appropriate client in its terminal emulator. It's not an SSH client that supports Telnet, and its terminal emulator doesn't do SSH on its own (e.g. mintty is based on PuTTY's terminal emulator, without the SSH client).