Analog signal just means the signal that you get from a microphone and cable.
Think digital = on the computer and analog = outside of the computer.
So naturally, to get the sound from the microphone onto your computer, you need some sort of device to convert it from analog to digital.
Many computer sound cards have a built-in 1/8in line-in jack. This essentially is an analog to digital converter, but it isn't of the same quality as a dedicated device and doesn't have a microphone preamp (I'm just a hobbyist and I don't really know a lot about preamps other than they amplify the sound and that they can differ in quality).
In addition, the other thing you get from an analog/digital converter (or interface) is multiple digital tracks at the same time. For instance, I have a Presonus Firepod (a pretty affordable interface). It has 8 mic/line inputs and 8 mic preamps. So I set up 8 mics on my drum kit, plug them into the interface, and then the interface has a firewire cable that connects to my laptop (interfaces also commonly use USB2).
The interface gives me 8 simultaneous audio tracks to work with in a DAW (it stands for digital audio workstation. It means ProTools or the other software suite of your choice. Happens to be Logic Express in my case).
I do all my mixing within the DAW, and I don't really understand why anyone would need a physical mixing board these days other than the fact that they are old fashioned and like having a bunch of knobs and sliders in front of them. Ha! But seriously though, I'm only an amateur and I'm sure there are plenty of reasons that the pros use them in digital recording setups.
Anyways, I think what Mediocrefunkybeat meant about A/D converters is that if you want to integrate an analog mixer into your setup, you would need an interface that takes output tracks from your computer, runs them through the mixer and then back through another interface (or the same interface if it has that many ins and outs) to get back into the DAW.
It's definitely more expensive and complicated to do. I wonder if there is someone with more recording expertise that would know why you might want to do this (other than to have physical controls at your fingertips rather than a UI on a computer screen)?