They work like gradients in photoshop, but with somewhat different tools and application. Each time you apply one, you can adjust several qualities within the range of space you've set up for the filter, and it will apply at a graduated intensity from the direction you first 'pulled on' the filter. The qualities you can adjust are: exposure, brightness, contrast, saturation, clarity, sharpness, and you can also apply a color. So if you want to bring in some more color in a sky, or take out some perhaps distracting colors in the bottom of your frame, or perhaps you didn't quite get the exposure right but not necessarily for the entire shot. Or maybe you just want to play, be creative, make a filmy light leak effect or some fun color gradients to create some magic...graduated filters are your friend! Here are a few screen shots to give you some idea, but the best way to learn is to play with it yourself, so get to it!
In the two screen shots below, the filter I am working with is indicated by the black dot with the white circle around it and the lines coming out from it. I am applying a creamy yellow graduated filter from the bottom up, at a slight angle. It is also adding more contrast to that area. In the second screen shot you can see the difference as I increase the sharpness to the area. This is my favorite use of filters besides the color, the ability to add a selective and gradual sharpness. There is another way to do this in LR, the adjustment brush is the 5th icon, right next to the graduated filter icon, but I find it more cumbersome and less easy to apply evenly. I'm just not as skilled with that tool, but it is also very useful!
The screen shot below shows how to add a color graduated filter. If you click the little rectangle you can select any color from that palette, at any saturation level in the entire spectrum. As you can see below that, I eventually chose a green to give some more blues and greens from the top down. You can see how simple and intuitive lightroom makes this process. Just try it out with extreme values and you'll catch on quickly to how it works!
In the screen shot below I'm adding a purple graduated filter from the top left, and I've decreased the brightness quite a bit. Those dots are what appears when you click on an area of a photo to start a filter. You can see there are three other filters in this shot.
In the screen shot below, I am adding some warmth, some pinkish color, and increasing exposure from the top down.
In this screen shot below, I'm adding a pink color as well as increasing general saturation in the area on the right, moving left across the photo. Therefore, the saturation of colors in the image is at the highest on the right, and gradually becomes less moving to the left.
In the screen shot below, I'm working with that filter indicated by the dot on the bottom. As you can see, this filter's main purpose is to decrease saturation, coming in from the left. So from the farthest left area has the least saturation, and saturation gradually increases from -45 as you move right across the image. Before I had done that, the yellows on that side overwhelmed the image and distracted from the overall softness of the shot.
Thanks for giving me this opportunity to show you a few things. I love teaching others what I know and helping fellow photographers to learn! Let me know if you have questions.