There several different styles of key extractors and the methods to use them vary a bit, depending on the type, but mainly they all involve catching or grabbing the end of the broken key with a sharp bit on the end of the extractor and then just pulling it out.
Some common ones are: spiral, saw and hook (it looks kind of like a harpoon and is the type that they put into most commercial pick sets to "pad" it and make you feel like you're getting more tools). I must have over dozen of those harpoon-looking extractors that I've accumulated over the years, and I rarely use them. Mostly I just grind them down into different custom pick shapes. HPC also makes a variation of the hook extractor in their 2000 pick series. They are a little more slender than the regular ones so they fit into keyways better and have four different variations of the shape. Peterson makes some unique extractors, including three different types of extractor pliers which are most commonly used on automotive locks. They also make a different variation of the saw extractor, and another type which they call a shim extractor that slides into the keyway along the side of the key and has a hook shaped cutout in the edge that digs into the side of the key. They also make a couple of different hook extractors that are very similar to the HPC 2000 extractors.
The kind of extractor that I use the most often is the spiral type, which you just thread into the keyway alongside the broken key, then pull. I have three different sizes that I use, depending on how much space is available in the keyway: .040", .060" & .074". In my area I run across Kwikset, Weiser and Schlage locks most often which all have fairly loose keyways so I typically use the .060" tip.
I have an HPC EZ-6 extractor kit in my main pick case that I usually carry with me, it has three different tips: saw, .040" spiral and .060" spiral (pictured below), and the tips can be stored in the handle. I like it because it has a large handle to grab onto, and it's the only extractor I've found so far that has the .060" tip. Most of the spiral extractors I've found are only available with the .040" tip.
I'm sure there's something I'm forgetting to mention here, but hopefully this answers your question.
