Just a high level overview here!
So, you want to make you own digital audience concert recordings. Where to start?!? In order to create a digital audience recording, we require the following:
 power to the mics
 analog-to-digital converter (ADC)
 storage device
There's some additional miscellanous stuff we'll need - like cables, adapters, mic stand, etc. - but we'll skip that stuff for now. If you plan on making soundboard recordings instead, I recommend reading through this anyway to develop a solid foundation of understanding. So...a bit more on each of the above 5 components required to create a digital audience recording...
We all basically know we need mics, right? This is a huge topic in and of itself which we won't get into right now. For now, let's just say: we all understand we need mics to create an audience recording. power to the mics
All condensor mics (the kind of microphones we use for our recording) require some kind of power to operate. Most condensor mics operate using phantom power. Phantom power is just a way of saying the mics are powered by an external power source. Just about all the common condensor mics you see tapers using operate with phantom power. The alternative is internal power. Some mics work with either phantom or internal power, two common examples: AKG C1000S and Nak CM100/300 series. gain
So we have our mics. And we have power to our mics. But the signal coming from those mics is usually very weak/soft/quiet. We typically need to apply gain. Gain simply strengthens the signal so it's strong enough to feed into an analog-to-digital converter. If the signal isn't strong enough, the quality of our recordings suffer. Now that we have a strong enough signal, on to the ADC... analog-to-digital converter (ADC)
The ADC translates into the digital domain (0s and 1s) the analog signal coming from our powered microphones and through our gain stage. Once the signal has been translated from analog to digital, then all we have to do is store those 0s and 1s somewhere. storage device
So where do we store all those 0s and 1s that are translated from analog by the ADC? Most common these days is a DAT recorder. Other common options include laptops, mini-disc, and hard driver recorders (a la Nomad JB3). Pretty simple right?
Do we really need 5 different components to perform these 5 different functions?
Sounds expensive! Well, not really. There are plenty of options for gear that perform one or more functions [1-5] above. For example:AKG C1000S > Sony D8
In this example, our microphones (AKG C1000S)  are powered by an internal battery . Likewise, the Sony D8 provides multiple functions: gain , ADC , and storage . Two pieces of gear (remember, we're skipping the cabling and such for now)...that's it!
Why do so many people schlep around so much gear if it's possible to create a digital audience recording using just a pair of internally powered mics and a portable DAT/MD recorder?
Usually, manufacturers who create devices which perform more than one or two functions [1-5] have to skimp on quality to keep costs down. Take the Sony D8 portable DAT recorder: in order to keep the costs down on the D8, Sony skimped on the quality of the preamp  and ADC . As a result, many people use an external gain stage  and ADC .
What about soundboard recordings, how do I make one?
Same principle as audience recordings, though we no longer need [1-2]. Note we still need  in order to control our recording levels, and we obviously still need [4-5] to convert the recording from analog to digital and store the recording. Fortunately, most storage devices include [3-5] in their functionality so often times the only required device is a storage device.
What kind of equipment do people usually mix and match when putting together their taping gear?
Lots of options!
Check out the thread titled I have a budget of $NNN - what gear should I buy?
for information on what gear fulfills which function [1-5].