doug has it right, but i'm going to add just a bit more here.
before digital, the resolution of something like a photograph was most simply defined as how much detail can you see in the photograph. and if you think of newspapers and especially older ones, you might remember that you could see the individual dots of an image in a newspaper. they were closely enough spaced that when viewed altogether you would see a picture. that's resolution. how many dots per inch were there in that newspaper picture was its resolution. a lot of dots gave you better or higher resolution because you would see more detail. fewer dots and it didnt look as good.
well, the same is true with digital. and there's a formula computer monitor manufacturers use to figure out what their monitor's maximum resolution is. i dont recall the exact formula, but it's along the lines of how many distinct dots can a human eye at 20/20 see from a certain distance away from the monitor. and that's the resolution. only, like doug says, when you're talking about computers and monitors, it's pixels. this is short for 'picture elements = pic els= pix els= pixels'.
ok, stop here if you dont want to go to sleep
but also like doug says, this can get a bit goofy. some folks will define the resolution by the picture dimensions, like 1024 x 768, or 800 x 600 and they'll call that the resolution. in fact, if you have a pc, that's how your desktop resolution is defined. and that just seems wrong, but it is also basically used, so it's become sort of right in that instance.
in photoshop and paint shop pro, there is a default resolution set at 72 pixels per inch. this has become a standard. but, you can change it in those programs. the reason i mention this is, you're going to hear folks say 'my resolution is 1024 x 768'. when they say that, they're usually referring to their desktop settings or their video card settings. when they talk pixel resolution they're usually referring to how many pixels per inch are in the image. so, you could have a picture dimension size of 1024 x 768 but not have it at 72 pixels per inch. what the graphic editors do is keep the pixels per inch stable (unless you force a change) and then alter the dimensions of the picture to fit that. but since your monitor ONLY wants to display a certain dimension, they use the picture dimensions as the stable factor and alter the pixels within.
sounds a bit nutty, doesnt it. but, you can try this for yourself. your desktop picture is nothing but a picture. you can actually load it into your paint program and watch the dimensions change when it gets loaded. and likewise, you can take a picture and load it as your desktop and watch those dimensions change. the reason i know this is, i took a HUGE satellite image i'd downloaded and made it my desktop and it ALL fit within the desktop. but, in my paint program it went way beyond the edges of the program window.
so, in effect, the desktop resolution IS defined by the length and width of the image, while in the paint programs it's defined by how many pixels per inch there are in the image. the bottom line, however, is still 'how much detail can you see in the image'.
Craig