Headphoneguide

Introduction
Why Headphones
Important things to consider before purchasing headphones
Budget segment up to $100 (no amplifier needed, though you always benefit from one)
Entry segment from $100 to $200 (from here on out an amplifier is recommended)
Midrange segment from $200 to $300
High end segment from $300 to $500
Audiophile segment from $500 to $1000
Audio purist segment at $1000+
Wired headsets, that are okay for their money, well designed and sound alright:
Budget DAC/AMP combinations (all portable in this category)
Midrange to high end DAC/AMP combinations

Introduction

Audio is vital, if not essential to a quality gaming, movie or musical experience. With a pair of great headphones your entire PC experience will be truly glorious!

Why Headphones

Headphones have the power to create the most immersive gaming, movie and audio experience you can possibly find. Always make sure to give the headphones you like a listen before purchasing. It's a very personal experience and finding the sound you like is important.

Here's a great guide on how to look at reviews: Understanding Headphone Reviews by Daniel Lu.

Headphones > Headsets
When you buy a headset, effectively you're buying a pre-build headphone + mic. This means, as always with pre-builds, that you are not getting the best pound for pound option. While some prefer this, you always have to realize you can get a much better sound by simply purchasing good headphones with a seperate microphone.
Hereâ€™s three videos of Tek Syndicate explaining why you always overpay with headsets and should stick with headphones and a microphone:
Gaming Headsets Suck - Make Your Own For $50,00 Or Less
Gaming Headsets Suck: 150$ Replacement Guide
Make Your Own Gaming Headset - $200-$1000 (High End)

Open headphones vs. closed headphones
Closed back headphones: isolated listening experience, great for making sure no one else is listening in, the sound is directly placed in your head (as if you're imagining the sound).
Open back headphones: open listening experience, as if you're in the middle of the scene/game/musical experience, sound can flow freely and others will hear it as well.

The most important things to take in consideration is whether you don't mind that others can listen in on what you're listening to (open) and if you do whether you like having the isolated, directly "in your head" type of sound experience (closed).

Impedance
To not get overly technical here, what this means is the higher the impedance, the more power is needed to drive said headphone. For example a 15 Ohm pair of headphones can be driven by virtually any audio gear, a 250 Ohm pair of headphones need an amplifier to be able to reach their real power. Keep in mind that when you have a low impedance headphone, powering them too high might result in a blow out, because they don't need much power to sound the loudest they can.

DACs/Amplifiers/Onboard sound
DAC: a digital-to-analog converter. A DAC is needed to convert the digital signal to analog to drive any headphone, speaker or amplifier in order to produce sound (analog air pressure waves). A DAC is in every soundcard, including onboard, the quality is the only difference.
Amplifier: an AMP is an electronic device that increases the power of a signal. This will effectively increase the volume, the quality of it determines how well (and with how little distortion). This is also the way to drive higher impedance headphones or get more sound out of lower impedance ones.

To truly get the most out of your headphones, a great DAC/AMP is always recommended. This doesn't go just for audio either, it will revamp your gaming and movie experience as well (louder, less noise/distortion, better quality audio).

In recent years onboard soundcards have improved quite significantly. For example the ALC892, ALC898 and the new ALC1150 codecs are great DACs and in fact you'd need one hell of a soundcard or DAC to even come close. ASRock and Gigabyte have gone a step further with the ALC1150 and actually implemented a 600 Ohm capable Amplifier in most of their midrange and high end boards. If you have a Purity/Purity 2 or Gigabyte system with the ALC1150, a budget DAC/AMP will not improve your experience as the headphone amp is already more than capable! However when you go to the higher segments of DAC/AMPs the improvement is still significant, no matter how you look at it.

Here's a Techquickie by Linus for a bit more explanation: Sound Cards, DACs, and Amps as Fast As Possible