I'm going to disagree with some of the comments here, especially the smallest pixel size comment.
Unlike CRTs, LCD monitors have a "Native Resolution", typically between 1280x1024 and 1920x1200 depending on how much you pay and whether you got a widescreen one or not. They look like crap at any other resolution.
So if at the native resolution the text is too small for you to read, it's going to cause you problems. I am absolutely sick of going into customer sites and having them complain about the picture quality of our imaging software only to find they're running a 1280x1024 LCD at 1024x768. So I fix the problem by fixing their resolution and then they complain that the text is too small! On the flip side, I managed to get us a contract supplying Dell LCDs to a company at a premium, even though we're more expensive than buying them direct from Dell, because the ones we supplied "Looked much clearer" than the Dell ones. The only difference is that I turn up, plug them in and set their computer to 1280x1024. They're the exact same LCDs!
It's made even worse by the fact that if you enable large fonts in Windows XP (or earlier) many applications don't display correctly any more because they were written by idiots who only tested them under small fonts.
So, my advice, go to somewhere like PC World (for god's sake don't actually buy anything) and have a look at their computer systems, make a note of the screen size and the resolution it's set to until you find a combination that you like the look of, then go out and buy a screen of that size.
The next thing to consider is what type of panel you want to get. This depends on what's important to you. The most common factors that bother people are viewing angles and response time.
Viewing angle is simple, it's how far to one side you can get and still see the image. TN-film panels have the worst viewing angles. IPS panels tend to be a lot better.
Response time is more complicated, basically it's down to the delay between the computer requesting that the picture change and the LCD actually doing that. Even the slowest LCDs these days are in the 60ms range which is absolutely fine for normal internet/email/word processing type usage. However if you play games at a competitive level then this delay puts you at a major disadvantage (however if you play games at a competitive level I'd expect you to already know this!). Predictably TN-film panels have the best response times, IPS not so good.
The last thing that bothers some people is the finish on the front of the panel, some are glossy some are matte. Glossy gives more vibrant colours. Matte is, well, less reflective. Everyone has their own preferences. I prefer matte as I am sat here with a window behind me with the sun shining on the monitor. The matte finish makes this much less annoying than my old CRT.
As for what to get. Have a look at the Dell range. They make some really nice LCDs and they're usually quite cheap. They also regularly run special offers. Alternatively go back to PC World, try out their LCDs, then buy the one you like from somewhere else!
I'm currently using a 26" Hazro LCD. These are nice monitors, though not the cheapest by any means. They're made in the UK too apart from the panel itself which is a H-IPS panel made my LG. The response times are incredible for an IPS panel at around something like 4ms and the viewing angle is pretty damn good too. Plus the all aluminium casing is very stylish if you care about such things. It seems to be the ideal compromise between being good for gaming and still being able to have people look at what you're doing (I suspect this may be the other reason why gamers like TN panels
)
If you can tell me what you'll be using your computer for I can probably come up with a more specific recommendation.
As for the DVI issue. Some of the stuff above is overcomplicating things a bit. Basically there are two types of DVI connector that you will encounter on an LCD monitor. DVI-I and DVI-D. the -D variant is digital only, and will require a DVI connector on your computer in order to work. The -I supports both digital and analogue so will work with the older 15pin VGA connector on your computer. Be aware that if you use an analogue path then your display will be being converted from digital to analogue and back to digital again, and picture quality will suffer as a result. If you're not using your computer for gaming then you can get a suitable graphics card for about 30 quid.
Edit: If you're planning to buy a screen with a resolution higher than 1920x1200 then you'll need to ensure your graphics card supports "Dual-link" DVI, and forget trying to use it with a VGA connector as it will look awful. However I doubt this will be an issue for you.