It can be difficult to know exactly what kind of headphone is right for you. You're probably choosing headphones because they have a better sound quality than earbuds. Only chose earbuds if you lack space to store them. I think an exception is the iPod. Some of them are really good. But if you don't have an iPod or you want the best quality the best choice is clearly headphones.
First of all, you want to look at the weight. You just can't listen to good music with heavy headphones. It's really uncomfortable and annoying. So the lighter, the better. To be more comfortable you may also want luminous earphone. A wire can really be disturbing if you're running, walking or moving to reach some files.
Of course, the sound quality of sound is also very important. Cheap headphones are usually artificially boosted to make the sound better. As a result, the sound will be significantly worse if you want to turn the volume up. For that reason, you can't find good noise cancelling headphones for less than 40 dollars.
The quality of the bass is also very important if you're listening to rap or hard rock. Most of the headphones are review had average or bad bass quality. So make sure you read a lot of reviews before you buy your headphones. Because even some 100 dollars headphones have a bad bass. However, remember that the sound quality is great for most 100 dollars headphones if you want to listen classical music or jazz.
If you want to run with them, you also want headphones that won't fall out during your workout. Behind the neck headphones are an excellent solution to avoid that kind of problems. However, it can be a problem if you're wearing a helmet or if you have a particular hairstyle. In that case, quality "clip-on" or neckband headphones can be a good solution.
Sport headphones are great if you're running on a treadmill for example. But they can be dangerous if you have to be aware of cars or bikes. So only buy sport earphones if you're running indoor.
I think the most important things are sound quality and durability. If you're on a tight budget that' the two things you want. If you have less than 40 dollars to buy your headphones then it's probably better to buy the best earbuds. Make sure to read reviews before you buy to be sure you make the good choice.