Having just taken my 65th flight for this year, and a few more left to go, I'm a bit of a FF veteran.
You essentially have three main schemes which are:
One World (BA, Cathay, Finnair, Qantas etc)
Star Alliance (BMI, Asiana, Singapore, Thai etc)
Skyteam (KLM, Air France, Korean, Delta etc).
When you are registered with one airline within the scheme, you can claim 'miles' on any airline within the alliance.
Then you have many middle eastern airlines who have their own seperate frequent flyer schemes, which occasionally team up with other airline partners.
When it comes to the miles, most airlines offer you types 'status' miles and 'reward' miles for each sector.
Status Miles
These tend to be time limited, and cannot be redeemed. They accumulate to give you access to a tier of status, which may give you benefits such as lounge access when flying economy, increased chance of upgrades, extra baggage allowance, priority check-in.
Reward Miles
These tend to be non-expiring, and once you accumulate these you can use them to book flights, normally paying just taxes and fees. Or you can use them to upgrade from economy to business.
One of the best schemes for UK citizens is the British Airways Avios Miles (formerly Executive Club), where you can accumulate miles from alternative sources including Tesco Clubcard miles. For members of this forum, you could use these miles towards Cathay Pacific flights.
If you fly once or twice a year in economy, it probably isn't worth having any loyalty to an airline, and you should book a flight based on a balance of your budget vs convenient flight times vs quality.
For those that travel more, or travel in business class (which accumulates more miles), then it is well worth factoring loyalty into your decision process for choosing an airline.
At the moment I hold Gold with KLM, Silver with Etihad and Silver with BA. KLM probably offers me the most benefits.
Sometimes the taxes and fees can be quite expensive, but can be useful if you need to book at the last minute when flight prices are higher.