You don't really need me to explain to you what headphones are, so let me give you some specifics that will help you decide which ones are right for you and your specific needs.
First, if you are going to use headphones for mobile dj or club dj use, you want to get a set of headphones that cover your ears completely. This type of headphone will block out as much of the external sound of the PA system as possible and let you hear the beats more clearly, so you can execute a great mix. Often times these are called "closed-cup" headphones. Some great examples of these include the HD series of headphones from Sennheiser, the ATH-M20, ATH-M30 and ATH-M50 from Audio Technica, and a variety of offerings from Denon and American DJ.
Next, you need to make sure that they are comfortable enough to be worn for your entire set, or recording session. I can't tell you how many people say they love the sound quality of their headphones, but they hurt after ten minutes. This is no good! Over time, I have found that most of our customers who don't want to spend hundreds of dollars on headphones typically gravitate to the Sennheiser HD280 Pro. At around $100.00, these are easily the best in class. For a little step up in price, we also find that our customers love the Audio Technica ATH-M50, which you can purchase in either white or black.
Headphones can also keep your hearing intact. After having done pro audio work for over twenty years, I am often asked why my hearing is still so good. The answer is simple, I wear my headphones all the time. I might not have the headphones turned on, but I have them covering my ears always. You can't put a price on your hearing, so I suggest you find some headphones that are comfortable and easy to wear for extended periods of time.
Next come the earbuds, these are the headphones that you literally stick into your ear. These are very affordable and they sound great. You will often see these used by people who are biking, running or jogging. The fact that they are small and compact makes them great for travel too, as they take up almost no space whatsoever. Most of the nicer sets come with different sized earbuds to fit you perfectly. I know mine came with four different sizes. Just about all headphone manufacturers offer some earbud type headphones.
Lastly, let us take a look at noise-cancelling headphones. These are a marvel of modern audio technology. Basically, they have a microphone which measures the sound in your environment and then creates a negative wave for those frequencies. This cancels out the exterior noise. Think of your elementary math class, the sound outside the headphones is +2, the electronics within the headphones then generate a signal that is -2, and so they cancel each other out. You often see these worn by people on airplanes or in other loud places. Audio Technica and Sennheiser make excellent noise-cancelling headphones.