The following is what I see as a simple guideline on what you should expect from an amp. Read through it and try to answer the question in the end before deciding whether an amp is indeed what you are looking for as the missing chain in searching for better sound quality.
A portable amp serves two general functions: 1) To replace the standard, lower in quality amplifying circuit in most portable music player in order to improve the overall sound quality. and 2) To provide enough 'juice' to audiophile's headphones which usually are much more demanding than most music player can handle. There are other purposes for using an amp, but I consider them to be minor and we won't discuss them here.
So why do we want to replace the internal amplification of our portable player? It is because most of them are designed for portability in mind thus the circuit must be compensated in both the size and material, resulting in an amplification circuit that are usually only good enough for basic task but inadequate for producing high quality sound. To totally avoid the internal amplification circuit, the usual method is to re-route the signal from the DAC (Digital-to-Analog Converter, the chip that converts digital data to analog sound before the amplification takes place) to an external amplifier via a line-out port/jack/dock (i.e. the line-out-dock, or LOD, for iPod user). A real line-out signal is what known as the 'purest', unmodified, uncolored analog signal. But the fact is, most if not all portable player on the market doesn't really have a real line-out. Instead, they deploy a pseudo line-out by re-routing an signal from the amplifying circuit in maximum volume, mimicking the stronger, higher output power of a real line-out. For those who don't know about it, one of those player is your regular iPod (*see note 1). This is why a few years back, a group of people start to modify iPod and sell it as 'iMod', an iPod that has its real line-out signal re-routed to its port so the user can use it with an external amplifier to avoid the what they believe to be an unacceptable internal amplification circuit. As for those who can't afford the high price tag of the iMod, you just have to stick to the so called 'tempted' pseudo line-out of your iPod. Essentially you are just feeding an glorified headphone-out signal in close to maximum volume / output power to your portable amp. You often find people commenting that external amp must be used with a line-out signal or there will be no point - of course, they seldom consider the fact that their line-out signal isn't a real line-out at all. Instead, the tendency for audiophiles taking their idea to a religious level often blind them from seeing the truth. You as the reader must learn the simple fact that audiophiles, including me, are not always correct and you shall never take our advice as it is.
Now we move to the second objective for portable amplification: Providing more power to drive the high impedance, low sensitivity headphone (or more relevant to our discussion, IEM and earbuds). As we discussed before, portable player are generally compensated in their internal amplification, which means the power they can provide is very limited and unsuitable for difficult-to-drive earphones. In the audiophiles' world, we like to say that the earphone is 'underpowered'. The chance is, most of you who are reading this article probably don't have an high impedance, low sensitivity earphone that really required an amplifier to sound at its best. Many of you just want to get an amp because you got advice somewhere from someone who claims that portable amp is the best thing that happens in the portable player's world and you are not hearing high quality music unless you put an amp to the back of your portable player. What actually is closer to the truth is, amplifier is one of those thing that generally add very little value to the overall sound. It is the kind of thing you want to pay attention to until you got your source and headphone right so they can be benefited by amplification. If you are still using a sub$50 or $150 (or even a $300) IEM or earbuds, the chance is you will not notice too much improvement from amping, at least not really worth the hundreds of dollars you paid for a well known portable amp currently in the market. I have seen people using easy to drive IEM like the Shure SE530 or the Ultimate Ears Triple.fi 10 Pro with iPod and an amp even bigger than the iPod on-the-go. For your average users, the setup might seen unpractical and often it is. Most IEM in the market are very sensitivity and easy drove by an portable player (and usually it is how they are designed to be), adding an amp to squeeze out the last 3% to 5% of sound quality should be the least of your concern when you can get more obvious improvement from headphone or source upgrade (or even by getting better music bitrates). You should consider an amp when you know your headphone is difficult to drive (which will benefit most from amping), or else the return will not be as dramatic as you wish it to be.
Is it ideal to amp signal from headphone-out? Probably not, but one has to make do with the equipments at hands. Is it really important to amp a sensitive earphone even though it is fine by headphone-out? Definitely not, but I can assure you some people will think differently. The art of balancing portability, practicality and music enjoyment is what important here. You must ask yourself one simple questions: 'Is it really necessary?'
If you still can't figure out the answer at this point, I'll suggest you start by using a smaller, cheaper amp. Either a Cmoy amp from eBay or something like a FiiO E5 should keep your wallet intact while providing a taste of portable amp. Just remember: Music enjoyment is from the heart, not from the gears.
Note 1: Post 5.5G iPod models do have a direct line-out (via LOD) based on Apple's own DAC chip. It is still arguable whether it sounds better than iMod, but it is said to be the best , clearest line-out signal across all generation of iPod.