The PlayStation Vita has sold less than 10 million units worldwide. All in itself that might sound like a pretty decent amount, but it's dwarfed when we consider what the normal numbers for handheld consoles actually are: The PlayStation Portable sold about 82 million units, Vita's biggest competitor the Nintendo 3DS has sold 58 million units, and the original Nintendo DS a whopping 154 million units. In this light the Vita, which has sold a tenth of its competitors, and even its own predecessor, is arguably a failure.
And that's not all. The system is arguably also a failure, even a bigger one, in terms of its game library. The game library for the Vita is pitiably small. And we are talking about all games available for the system here. The triple-A game library for the system is significantly smaller still, to an almost ridiculous level. It's hard to sell a console that has no games for it. And this is a vicious circle because developers won't make games for a system that doesn't sell.
Why did it fail so catastrophically? Many people agree that this was caused mostly by two factors: Sony's greed (which is easy to believe) and, perhaps surprisingly and unintuitively, because the system is actually too powerful for its own good.
The first one is much easier to explain: Not only was the console quite expensive at launch, on top of that Sony got really greedy and crippled it with a proprietary memory card, which was over twice as expensive as the normal generic ones. What is worse, the Vita was shipped without a memory card (at least one that was large enough to download any games from the PlayStation Store), which meant that a separate memory card purchase was pretty much required to be able to use the system, which increased the actual price of the console even further. (This was, in fact, a rather dirty tactic from Sony. Not only was the launch price of the system quite high, on top of that it was actually artificially and deceptively lowered by not including a necessary component, which you had to then buy separately. In other words, there was a hidden cost, which wasn't very small either.)
Even the smallest of the proprietary memory cards costed like 20€, but it was so small as to be barely enough. If you really wanted one that you could actually use for actual games purchased digitally, you would easily end up paying 40 or 50€. On top of the original unit's price, of course. (At the same time, standard memory cards of the same capacity by other manufacturers would cost less than half of that. And there is no technical reason why they couldn't work on the Vita, except Sony's greed.)
The second reason may be harder to fathom at first, but let me explain.
The Vita contains some impressive hardware for a hand-held. It is, of course, hard to make comparisons, but it has approximately the same prowess as a PlayStation 3. On a handheld. And not only is the CPU and GPU so powerful, the screen is quite impressive in itself, being large and high-resolution (for a handheld at least).
But how can being "too powerful" be a detriment for a handheld console?
The reason is that making games for a powerful console is more expensive. And with the Vita, it's a gamble. A game studio may spend millions creating a Vita game, and see it sell only a fraction of what's necessary to cover the costs.
Nowadays most smartphones are about as powerful (if not even more so) and have even higher screen resolutions (even ridiculously so), yet they are quite successful. How come? Well, cellphones are not competing on the exclusive market of video gaming. A cellphone is not a dedicated gaming console. It's a smartphone; not just a phone, but essentially a portable mini-computer which you can use to do all kinds of things (such as browse the internet, message with people, and use all kinds of apps and games.)
The Vita, however, is just that: A gaming console. That's its principal purpose. Sure, you may be able to surf the internet with it, but nobody uses it for that purpose. It's not a smartphone.
And a gaming console needs a healthy library of games, or else it won't succeed.
The Vita has thus entered a vicious cycle from which it can't get out: It's too expensive and too risky to make big triple-A games for it, which means that its game library is very small, which means that people won't buy the console, which means that game studios won't make games for a console that doesn't sell... and so on.
It didn't help that Sony got greedy about it. Maybe if they hadn't been so greedy, it would have been a different story, even with its current hardware prowess and subsequently increased development costs. But they were, and this happened.