Ariel compressors are the gold standard around here for reciprocating gas compression, if your in North America and use natural gas the odds are it went through at least one Ariel compressor along the way. Pretty cool they are built that close to you.
As has already been mentioned, in the most basic sense a natural gas compressor is like an air compressor that is completely sealed. Suck in gas at low pressure, compress the gas in one or more stages, and discharge it. Again as already been said this is done for gas transmission.
Gas compression is also needed for gas processing, I work in the upstream and midstream side of the industry, basically anywhere from the well to the long distance transmission line. Generally you don't have an oil well or a gas well, usually it's a combination of both with water, it's the proportions that change and denote the naming. A well will produce hydrocarbons ranging from wax or bitumen all the way to methane, with thousands of products in between.
Raw "natural gas" is the same way. Usually you will have from C1 to C4 (number of carbon atoms in the hydrogen carbon chain) in the mix. Quite often there are even traces of heavier C5+ compounds. When a gas is compressed these heavier molecules tend to condense into a liquid where they can be separated out. Generally it's possible to extract the majority of molecules heavier than C3, and even a good portion of the C3 with compression alone. The remaining C3 (propane and it's isomers that are sold as LPG) are more valuable than the methane, and there is financial advantages to separating this remaining propane out. In this case refrigeration is used to help condense out the propane, ironically using a compressor pumping propane in a sealed loop as the refrigerant, chilling the process gas to around -15c allowing the propane to condense and be separated out.
This leaves C1-C2 gas behind. C2 is ethane and it's isomers. Ethane again is more valuable than the methane, so there is advantages to stripping this out. It's also for safety of the consumer, natural gas usually runs about 1000 btu/cuft, leaving lots of ethane or propane in the gas increases this btu rating considerably, burning hotter and can even run a risk of overheating natural gas appliances. Again this is stripped out by compressing the gas and also cooling it, either with a separate refrigeration loop or the process gas itself, except this time the gas has to be cooled much lower to drop out the ethane. In a big plant this refrigeration step takes lots of energy, and a big compressor to run it.