Could you explain to me what this program does, and how you use it with Photoshop?
That's a good question and one that Adobe doesn't do enough to answer.
Lightroom has five modules - Library, Develop, Slideshow, Print, and Web. The Develop and Print modules handle things that Photoshop already does well, but Lightroom does them differently.
The Library module helps you organize and keep track of your pictures. You can group your pictures into collections, which can be nested. So you might have a Disney Collection and within that, you might have a collection for each trip. You can also apply keywords to your pictures. That allows you to search for your pictures that contain a particular person, place, thing, or whatever you keyword it with.
The Develop module is like Photoshop in that it is used to adjust the appearance of your pictures. It also does RAW conversions. It can be used to adjust the white balance, exposure, saturation, curves, and even some noise reduction, sharpening and lens correction. It doesn't have layers and only a few things (like cloning) can be applied to selective areas. You can't, for example, use it to increase the exposure for part of the picture but not all of the picture.
A key area in which it differs from Photoshop is that all of the adjustments are done with metadata. In Photoshop, when you change things, you are actually changing the pixels in the picture (or on a layer). In Lightroom, you are just adding metadata (instructions) on how to perform the change. That means that the files are smaller and you don't need to rely on layers to be able to back out changes.
I also like the Develop module because it helps streamline the adjustment process. I find it much quicker to make adjustments to pictures. In Photoshop, you have to open each picture, make your adjustments using different menu items and pallettes, and then save the picture. In lightroom, you just pick the next picture on the filmstrip, make your adjustments using the nicely laid out tool pallette, and then move on to the next one. Using Lightroom saves me lots of time.
It has a module for making slideshows, but I haven't messed with it much.
The module for printing is a nice start, but it doesn't have soft proofing. Soft proofing is the ability to have the program display a good estimate of what the picture will look like when you print it. Assuming that you have everything set up right (which is pretty complicated), you can use Photoshop to get a good estimate of how your prints will look. That gives you the chance to make adjustments to the photo before spending lots of money on paper and ink. Because of the lack of soft proofing, I haven't done much with Lightroom's print module.
Finally, it has a web module. This is for building photo websites. I haven't played with this much either.
For me, Lightroom is really just a photo library and photo processing program. It's pretty good as a library and great for rapid prototyping. If you only shoot a couple dozen shots a month, it probably isn't worth bothering with. If you shoot hundreds of shots a day when you're out shooting, it's almost indespensible.