There are around 1.6-1.7bn* PCs in use today, and there are already perhaps 2bn iOS and Android devices**. Over the next few years the great majority of the mobile base will convert to these devices: there will be 3-4bn smartphones in use*** and hundreds of millions more tablets.
So, mobile means there will be two to three times more personal computing devices connected to the internet. But actually, that understates the change massively. The difference in how those smartphones are used is actually just as important as the raw numbers.
First, they are not shared and they are personal. Of those 1.6-1.7 bn PCs, a little over half are consumer devices, and a large proportion of those are shared. The others are owned by companies, and at the very least they're restricted in what you can do with them for personal uses, and many of them are actually single-purpose devices. So it's helpful to think about somehow discounting that PC base to reflect actually personal personal computers - by half, or more. Just as there's a 'full-time equivalent', what's the 'personal computer equivalent'? It's not 1.6bn - it's probably more like half that.
Then, these new devices go everywhere. The British term 'mobile' is rather more helpful than the American 'cell phone' (or the German 'handy'). They take the internet to wherever you may be, so they're available across the whole day, and they're available in different contexts: 5 minutes using Amazon in a store may be worth more than an hour on the Amazon website at home (this alone significantly broadens the scope of industries affected by the internet).
Next, these devices are much more sophisticated than the 'web browser + mouse & keyboard' paradigm that's been the desktop internet for the past 20 years. The camera is arguably the one of the most important input methods, with perhaps 2bn photos shared every day already, and location, push notifications, motion sensors and everything else keeps accelerating the sophistication and richness of what's possible well beyond the desktop web.
Finally, the step change in ease of use provided by the new generation of operating systems changes what it means for someone to have such a device. A very large proportion of PC users would describe themselves as 'not computer literate', or at best getting by following 'recipes' within a narrow set of tasks, but far fewer say they're not phone literate or even smartphone literate (though a curve obviously remains). The usability of this new class of devices of itself multiplies the reach of the internet.
When you pull these strands together, smartphones don't just increase the size of the internet by 2x or 3x, but more like 5x or 10x. It's not just how many devices, but how different those devices are, that has the multiplier effect.
To put this another way, if you were to go from 50m text-based mainframe terminals to 100m PCs, you could say you've doubled the market, but that would miss the point - you've improved things by 10x, not 2x. The same applies to smartphones.
Finally, this change in scale is multiplied further by the general collapse in the costs of building a product and taking it to market over the past decade or so. Regardless of what you think of Yo as a product, one programmer got to 1m pretty active users in a month with no funding. How much would it have cost to support 1m active users of anything in 2000?
That is, mobile will expand the scale of the opportunity by 10x, while the cost to reach that opportunity, at least in theory, has decreased by at least 10x as well.
** Google reported "1bn" Android users at Google IO (so perhaps 1.1bn over the summer). Chinese Android is at least 400m and not included in Google's number. Trailing 24m iOS device sales are a little under 470m: taking a slightly longer life-span (which is certainly the case for iPads and very probably for iPhones given the second-hand market) would go to 600m, perhaps more.
*** There are over 6bn mobile connections, but this includes a very large amount of multiple SIM ownership and dormant connections. The GSMA estimated 3.4bn human users at the end of 2013, and that's at the low end of estimates.