Right I'll start off and then I'm sure others will chip in to fill in any gaps. I will caveat that my examples stand for road bikes but the principles hold true for all types of bikes.
The numbers relate to the number of teeth on the cogs at the back or the chainwheels at the front.
So, people may talk of running a 12-25 cassette. That means the smallest cog will have 12 teeth and the biggest 25. Depending on the number of cogs, your preferences and the manufacturer will vary the number of teeth in between but on a Shimano 10 speed it would typically be 12, 13, 14, 15, 16, 17, 19, 21, 23, 25.
The chainwheels obviously have more teeth. So people may talk of a 53-39 (53 being the bigger ring, obviously). This is a 'standard double'. Or there is the compact, often 50-34, or triples 50-39-30. These are standard sizes off the shelf sizes - it is possible to spec your own.
The gear you ride is often quoted front teeth first, rear teeth second eg: 53x12
Why does this matter? Well, the gearing you use depends on the kind of rider you are and what sort of riding you do. As well as your fitness level.
Lower gears - so more teeth at the back, fewer at the front - are good for climbing. In the above examples, a 30x25 would be the lowest gear on the triple, ideal for cycling the north face of the eiger.
Higher gears (eg: 53x12, which is very high) is probably only truly useful for long downhills and racing.
The important thing is that you can turn the gear comfortably without straining but not so easily you go nowhere fro lots of effort, it's about finding the optimum - which varies according to the person. The other point is some people prefer/need the skips in the gears to be minimal (big jumps in racing are bad for maintaining a rhythm) but for hilly sportives you need a mix of very low gears to get up the hills but also big gears fro the flat sections and the downhill.
So, when people say 'I wouldn't ride 53x12' They are (probably) saying that is too high a gear for them. If they say 'Iwouldn't ride a 12-25' they are saying that size cassette is not suitable for what they do. They might want an 11-21 for pure racing with very small gaps between the cogs, or they may want a 14-27 because they need the lower gears fro touring or getting up big mountains.
In summary big at the front small at the back means big gear ideal for racing. Small at front big at back, good for pedalling up hills. Alot of the 'I'd rather ride...' is about personal preference. One person may say '53-39 with an 11-21 is ridiculous, get a compact and a 12-27' but someone who races would need an 11-21 to survive in a race.
Hope that helps.