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CRTs: Very cheap, but they're very heavy, use a lot of electricity, generate a fair amount of heat, and have little to no resale value.
LCDs: More expensive, but light, energy-efficient, cooler, and retain their value better.
Beyond that, I just make sure to go with a name brand (quality) and look for a warranty (reliability). I look for warranties when I buy not because I expect to take advantage of it, but rather the opposite: Something that comes with a lengthy warranty is usually built to last.
You should always consider the pixel resolution. Windows notionally assigns 96 pixels per inch. As a rule of thumb, you don't want to stray too far from this value without a good reason. A 15" LCD 1024*768 monitor displays 85 pixels per inch - that's probably about as far as you should go for normal use unless you have poor eyesite or plan to position the monitor slightly further away than is normal.
For web design purposes, you also want a monitor that works at different resolutions. Modern LCDs make a reasonable fist of this, but they are not perfect. If possible, check this before purchasing.
Last, but not least, you should consider ambient lighting. Unless used with very soft ambient light, you should avoid glossy screens. The last time I looked, most LCD monitors were matt but glossy monitors were beginning to appear. Bear in mind that glossy screens are normally described as anti-reflective and matt screens are normally described as anti-glare - very confusing.
Kaled.