An audio interface is simply a small box that you can use to connect your microphones or instruments into to record directly to your computer.
The most basic ones will have a single input, for a microphone or line in instrument, and more advanced ones will have more inputs available.
They also have direct monitoring, so you can hear what you are recording in "real" time and listen back to what you recorded.
A mixer is another way to get sound into your computer, and they also have live performance capabilities. They offer you more control over each line that is put in (microphones and instruments) before they go into the computer. Because of this they can be used for live performances, or you can skip some of the mixing that you might do in your DAW.
Depending on your needs, there are small ones that have one or two inputs, or large ones with a lot of inputs available, but they are not really portable.
USB microphones have built-in interfaces that allow them to directly communicate with your computer so you don't have to purchase microphones and an interface separately.
This site is only for suggestions and examples on recording equipment and software and suggestions on how to set up your recording equipment, not exactly what you should buy and not exactly how to set things up. Depending on your space, equipment, and live/studio setting, you will have to experiment for the set-up that will produce your desired results. Don't be afraid to experiment and go outside of the suggestions laid out in this site.