Sound PA Desk / Multi-Channel Mixer
Presuming a live mixer for a small to medium-large setup, this should give you an idea of how to to buy a mixer. The main thing to avoid is buying something too small as upgrading within a year is expensive and a pain to refamiliarise.
Input Channels
Try to determine the most number of channels you are likely to need at any one time, eg. a Carol Service or full compliment of musicians. Now add at least 50% and you need to buy a mixer with roughly that many channels. Eg. if you came up with 6 the you need a mixer with 8 channels or more.
The reasons are simple, some channels die and on mixers the final few channels have less control than the first few (a kind of unwritten rule). Don't forget that some mixers call a stereo input a 'channels' but that won't have fully slider control.
Most small 4 channel mixers are -quite frankly- a bit rubbish. Ideally going for 8 channels or more is best as you'll get several extra rows of Equaliser and Monitor/Foldback Channels.
Sub Masters & Multiple Speakers
In most cases they jump in 4 channel sizes and once you reach 12 channels or more you should get some sub-mixers so you can control a group of channels together (eg. instruments seperate from voices).
This can make it easy to say take down the band together before they unplug but keep the others like the speaker up. A personal preference that makes it even easier is to have the channels split either side of the master and sub master sliders so you can visually split different types of channels (though don't have to).
Separately you may end up with a scenario where you have left/ right and centre speakers if not also a second separate set of left and right. This might be unnecessary in a portable setup but in a fixed scenario having a centre speaker for spoken voices and seperate output for say the lobby or parents creche can really help.
Auxillary / Foldback Channels
Having more 'other' outputs which you can use for foldbacks or recording can really help. One foldback-fits-all doesn't usully work, eg. the lead musician might just want to hear only themselves while the keyboardist needs a bit of lead and some backing singers too. The more channels you have the more options it gives you.
Ideally you don't use these in place of a multiple output channel as any changes you make to the main mix (eg. via sliders) won't be reflected in these sub-channels unless you make the same change. However to make a recording of a talk they are ideal as you want a constant input and only from a few channels.
Having more 'other' outputs which you can use for foldbacks or recording can really help. One foldback-fits-all doesn't usully work, eg. the lead musician might just want to hear only themselves while the keyboardist needs a bit of lead and some backing singers too. The more channels you have the more options it gives you.
Ideally you don't use these in place of a multiple output channel as any changes you make to the main mix (eg. via sliders) won't be reflected in these sub-channels unless you make the same change. However to make a recording of a talk they are ideal as you want a constant input and only from a few channels.
Other Stuff
Phantom Power is only useful if you have kit that actually needs it and that is pretty much very small microphones (such as those hanging above a choir) that need the power as they have a too small a diaphragm to make a signal from the input sound. On larger mixers you'll find it on some channels but by no means do you need it on all.
It can be nice to have LED indicators at the start of each channel, giving a visual indication that something is coming into the channel but beware that this can considerably increase the cost and isn't really useful unless you can't see all the potential sound sources.
For darker environments the larger desks sometimes come with light fittings which provide task lighting that you can move around. Although definitely a frill it can be a lot more practical than trying to use a normal lamp (eg. an anglepoise) and be constantly shielding the audience from it.
