The basic choice when it comes to printers is between an inkjet or a laser printer.
These are relatively cheap (under £50 for the cheapest models) and quite compact. As the name suggests, they work by shooting a jet of ink (or more accurately, tiny drops of ink) on to the paper. The position of the ink droplets can be controlled very accurately, so this gives clear, sharp results.
The big problem is that the ink is very expensive (several thousand pounds per litre!), so if you are doing a lot of printing the running costs are high. In some cases it can even be cheaper to buy a whole new printer than to buy a replacement ink cartridge (discussed here on Slashdot). It seems likely that the printer manufacturers subsidise the price of the printer, and try to make money on the ink cartridges.
They are also rather slow to print each page, and if the printer is not used regularly, the ink can dry up and clog the nozzles.
For these reasons I would not recommend an inkjet printer for an office setting. For home use, if you will only be printing relatively small numbers of pages, an inkjet might be the right choice.
Inkjets can give very good results if you are printing photos (sharp, and good colour), although you need to use special photographic paper (also very expensive) for best results. If you are thinking of buying an inkjet printer primarily for printing photos, do compare the costs with the cost of prints from a photo printing shop or online company — these can often be cheaper than printing yourself, if not so convenient.
If you go for an inkjet, look for a model which has separate ink cartridges for each colour (otherwise you might have to pay for a whole new cartridge when only one colour has run out). You don't have to buy ink cartridges from the printer manufacturer — you can usually get “compatible” or re-filled cartridges much cheaper, though the quality can vary!
These tend to be somwhat larger than inkjets and a bit more expensive. A basic colour laser will cost £250 or more; you can also get black and white only models for £150 or less. They use a dry powder toner (like a photocopier — they work in a very similar way) rather than ink, and this usually works out much cheaper than an inkjet in the long run. They’re typically much faster to print a large number of pages.
You can also buy "multi-function devices" which act as a scanner
and photocopier as well as a printer. Sometimes they are also a fax machine
(though not many people are still using fax machines these days). These
can be quite handy if you need the extra functions. They can be based on
either inkjet or laser printers, so all the above pros and cons apply.
One final thing: quite a few printers now have wireless access built in, so you don't need a cable to hook them up to the computer.