Matt's on the money when he suggests...
Influence is not the same as popularity
Influence is not the same as fame
Influence is not the same as the nature or form of our connectedness to each other (and by the way most human social networks are not of the hub-and-spoke variety that the Influentials Hypothesis proposes).
....though each of this can help us understand aspects of influence.
So what is it?
Influence is what makes the difference between an idea or behaviour being adopted (or not) amongst those around us and those around them (and so on).
Influence is often much easier to see after the fact and harder to predict ahead of time than we imagine - as Duncan and others have repeatedly pointed out - so we shouldn't imagine that because something does or doesn't seem to be a driver of others' behaviour that it would do so if we re-played the tapes).
But it's probably best understood as something the Influenced do in response to those around them (as opposed to something the Influential do to them).We call it Social Learning or plain old Copying
Influence is not fixed for most modern human life but fluid - not least because most human life now consists not of repeated interactions with the same few poorly connected individuals (beyond our little closed tribe) but rather of this plus fleeting indirect connections to millions of others and their groups and their connections. Not to mention the fact that most choices over which we permit other folk to have an influence are between equally good, fundamentally indistinguishable options (certainly compared to the few life-or-death decisions subsistence tribes face).
So next time the clowns at Klout start wittering on about influence, stop before you let them shape your opinions...