+1 for Audio-Technica M50S
An amp (short for amplifer) amplifies, or increases the power of an electric signal. Before your songs are produced as sound waves by your headphones, they go through some converting, processing, and amplification in your laptop or ipod. Here is a brief laid-back summary of the whole process, including an explanation of what the amp's purpose is.
First, the digital file that is stored on your device must go through a DAC (short for Digital-to-Analog Converter). What the DAC does is take the digital signals (1s and 0s binary code), and convert them into analog signals (variations in voltage, current, frequency, or electric charge). These signals are very weak and need to be boosted to a level at which they can be easily manipulated by mixers and equalizers This is done by the the preamplifier (or preamp for short). The preamp takes the weak analog signal produced by the DAC and boosts it to line-level. The preamp applies a variable amount of voltage gain to a signal, increasing how loud the signal will be after it passes through the power amp and through your headphones. Which brings us to our next component in the line-up, the power amplifier. The power amp provides the current gain to the signal coming from the preamp which is necessary to drive a pair of speakers or headphones.
Many times the preamp and power amp are bundled together in the same enclosure and referred to as an integrated amplifier. If you add in an AM/FM tuner, it is referred to as a stereo receivers (unless, of course, it supports more than 2 channels, then it would be a multi-channel receiver). Most of the time, when people talk about amps, they are talking about these integrated amplifiers, which include both a preamplifier and a power-amplifier.
If you are setting up a home entertainment system with bluray/dvd players, hdtvs, and a 5.1 or 7.1 channel surround system, you would typically purchase a multi-channel A/V receiver, which can take many, many audio or video signals (both analog and digital), and act as source selection, signal processing, and volume control for your whole system. An amplifier is part of receiver.
But, if you only listen to two loudspeakers and have no need for video hook-ups, you may prefer to have an audiophile-quality integrated amplifier, or even separate preamps and power amps. (Power output is as low as 30W to over 100W)
Or, if you listen only through headphones, you may want a nice headphone amplifier. Headphone amplifiers are made specifically with headphones in mind, designed to drive only them and not loudspeakers (There aren't any speaker-cable connections). Some come with 1/8" TRS inputs and outputs, made with iPod/PMP users in mind. The power output of these are usually only around 10mW to 2W (compare that to output of "normal" amps). Some even come with built in DACs, and are then referred to as DAC/Amps. Often times they include a battery so you can take it on the go. They range in portability and power source (AC power supply vs. Lithium-ion battery), some are small and less than 2 in x 2 in, and some are much bigger. The ones that are usually much bigger and heavier, and require an AC power source (plugged into a power outlet), and are referred to as Desktop Amps. The ones that are only 2" x 2" and have a battery lasting hours and hours are referred to as Portable Amps.
The Audio-Technica M50s are low impedance (38 ohms) and are driven just fine to a good volume with a laptop sound card or iPod. With these particular headphones, being driven by an ipod, and watching youtube vids, I personally would not purchase a separate amp for them. I am assuming you were looking at the portable amps by or similar to Fiio's offerings. If you don't want to spend much and just want to try out an amp to see what it offers, the Fiio E6 is pretty good for the price ($20 - $25/with cable), offering 3 different EQs (flat, and 2 different bass boost settings).
Hope I could help.
Edited by mpsipos - 7/15/12 at 11:29pm