I love air conditioners, I've had loads of them in my time and currently have two.
Air coolers are much, much cheaper, and they aren't so bulky or heavy to move around. They work (usually) by air being fanned over very cold water or ice cubes put into the machine. I don't like them because they're only like a glorified fan and when it's really hot, they just don't get the temperature down enough. They can also humidify the atmosphere which I personally don't want but some people find that helps them breathe in hot weather. I worry about it being too damp and producing mould. Coolers don't dehumidify (obviously) and don't purify the air. They are much easier to install however, as they don't need to have an outlet pipe.
Air conditioners are much more complicated and so cost a great deal more. They have a refrigeration unit built in which makes them extremely heavy to move around (eg. carry to an upstairs bedroom)...but in my opinion, worth it. The unit actually takes in hot air, sands it through the refrigeration unit and makes it incredibly cold, capturing the heat and directing it outside. To do this they need a pipe either slotted through a window or even through a hole in the wall, this limits where you can put them in a room. As they draw air in, they actually clean the air that's pumped back out again so if you have a few air conditioners going fairly constantly in the summer, you will get to a point where your air smells very fresh and 'clean' and you will notice a huge reduction in dust around the house. However, all this activity is costly, most average at about 50p per day on running costs, and they are noisy machines when running.
Air conditioners are measured in BTU and the higher the number, the larger the space they will cool. You can get a reasonably priced machine but then realise it's only 7000 BTU, which really won't do much in your home. I personally would go for a 10,000 BTU in an average bedroom and at least a 12,000 for a living room....you can, if you are prepared to pay for it, get a 15,000 BTU which is wonderful for a large room on hot summer days. Also check the specification details, some only cool the air down to 18 degrees but better ones take it down to 16 degrees. On really hot days you set your machine for 16 but it will never get down to that point, it's just that it will keep on cooling constantly, while it's trying, and you'll stay blissfully cool.
You can buy air conditioners online or in some shops, eg. Currys. They usually don't stock them until May onwards in the UK. Online, you'll often see "not available until May" on the listing. To get a bargain, spend time checking online for cheaper machines about now (Jan, Feb, March). These are last year's stock which they haven't been able to get rid of and can be £100 or even £200 less than they'll be when new stock comes in for the summer. (And by the time it's a July heatwave you won't be able to get one anywhere).
Both coolers and air conditioners last for years and need very little maintenance.
I hope that helps.