Analog soundboards are commonly found in theatres. They are technically outdated, but still perfectly useable, and some sound designers actually prefer them. In general, they have a much simpler learning curve than digital sound boards.
Analog sound boards have a slider and a set of knobs directly corresponding to each input (place you plug something in) on the board.
The slider, usually found on the bottom, controls the volume.
Above that, you'll usually find a mute button.
Next up is a knob to control the lowest pitched sounds.
Above that is a knob to control the mid-range low pitched sounds.
Then you'll find the knob for the mid-range high pitched sounds.
Then you'll find the knob for the highest pitched sounds.
Finally, you'll find a knob for the gain. This controls essentially how hard the input works to pull in sound. If you have a performer wearing a mic far from their mouth, or standing far from a mic, or even just with a quiet voice, you'll want to turn the gain up so it picks up their voice better. If the singer is near a mic or has a loud voice, you'll want to turn the gain down. For each microphone, you'll want to find the sweet spot with the gain where the sound can be heard audibly, but the sound system doesn't protest at all by "feeding back" (unpleasant noises), even at the loudest points. Once you've found that sweet spot, set it and forget it.
Every knob turns up by turning it to the right, and turns down by turning it to the left.
The main thing you'll need to be aware of after setting the gain is the volume sliders. You'll want to find out how high you can turn up each microphone with people singing or speaking their loudest. You'll have to play around with the volume as you rehearse or perform, to ensure that each performer can be heard audibly, but the system doesn't feed back. In an ideal world, you want every performer to be heard comfortably, but without their voice sounding unnaturally amplified.
The final thing you'll need to look for on the board is your "grand master". This is a slider or two (usually two) that controls the overall volume that goes out through the speakers (amplifiers, or "amps"). If this is turned down or muted, you won't hear anything through the speakers. Think of the board this way: use the sliders and knobs for each input (microphone) to "mix" the sound so that it's all evenly balanced exactly the way you want it, and then use the grand master to turn that overall mix up or down so that it's comfortable for the audience.
Be aware that sometimes there is a separate switch on a wall or console to turn on the amplifiers (speakers). If this isn't turned on, then you won't hear anything, no matter what you do with the board.
DIGITAL SOUND BOARD
Digital sound boards are newer technology, and thus much more expensive than analog sound boards. They offer some distinct advantages over analog boards, including a smaller size. Smaller size is often more convenient, and it can make operating the board much more comfortable.
The biggest difference between a digital board and an analog board is that digital boards don't have a physical input that directly corresponds to a spot on the board. Instead, you have to digitally assign each input to a specific spot on the board. This ends up being very convenient once everything is set up, but that initial setup process can take some time, especially with a board or situation or operator that is new.
In general, digital sound boards have a much steeper learning curve than analog boards. Once you master a board, it becomes very easy. But your average person can't just sit down and figure out how to run a digital board with no idea of what they're doing. You probably could with an analog board.
If you are planning to do a show with a board, situation, or operator that is new/different/unique, than make sure to plan plenty of time at the beginning of tech week for your board operator to work in a quiet, controlled environment so he/she can be sure to get everything set up properly. This will protect you from a myriad of problems and annoyances that you might otherwise have to fight all through the show.