What mistakes, if any, were made in Numberphile's proof that $1+2+3+\cdots=-1/12$? This is not a duplicate question because I am looking for an explanation directed to a general audience as to the mistakes (if any) in Numberphile's proof (reproduced below). (Numberphile is a YouTube channel devoted to pop-math and this particular video has garnered over 3m views.) 
By general audience, I mean the same sort of audience as the millions who watch Numberphile. (Which would mean, ideally, making little or no mention of things that a general audience will never have heard of - e.g. Riemann zeta functions, analytic continuations, Casimir forces; and avoiding tactics like appealing to the fact that physicists and other clever people use it in string theory, so therefore it must be correct.)
Numberphile's Proof.
The proof proceeds by evaluating each of the following:
$S_1 = 1 - 1 + 1 - 1 + 1 - 1 + \ldots$ 
$S_2 = 1 - 2 + 3 - 4 + \ldots $ 
$S = 1 + 2 + 3 + 4 + \ldots $ 
"Now the first one is really easy to evaluate ... You stop this at any point. If you stop it at an odd point, you're going to get the answer $1$. If you stop it at an even point, you get the answer $0$. Clearly, that's obvious, right? ... So what number are we going to attach to this infinite sum? Do we stop at an odd or an even point? We don't know, so we take the average of the two. So the answer's a half." 
Next:
$S_2 \ \ = 1 - 2 + 3 - 4 + \cdots$
$S_2 \ \ = \ \ \ \ \ \ \ 1 - 2 + 3 - 4 + \cdots$
Adding the above two lines, we get:
$2S_2 = 1 - 1 + 1 - 1 + \cdots$
Therefore, $2S_2=S_1=\frac{1}{2}$ and so $S_2=\frac{1}{4}$.
Finally, take
\begin{align}
S - S_2 & = 1 + 2 + 3 + 4 + \cdots 
         \\ & - (1 - 2 + 3 - 4 + \cdots)
         \\ & =   0 + 4 + 0 + 8 + \cdots 
         \\ & = 4 + 8 + 12 + \cdots 
         \\ & = 4S
\end{align}
Hence $-S_2=3S$ or $-\frac{1}{4}=3S$.
And so $S=-\frac{1}{12}$. $\blacksquare$
 A: The first (and fatal) error is the argument that $S_1=1/2$.  That's simply not what the definition of an infinite sum is; the definition is as a limit.  To show that $S_1=1/2$ (as opposed to not existing at all), you have to prove that the limit of the partial sums is $1/2$; you're not just allowed to make up whatever plausible-sounding argument you want.  (There are also similar errors in the computations of $S_2$ and $S$--those computations use manipulations that can be proven to be valid if you already know the sums converge to some number, but they cannot be used to prove that the sums do converge.)
However, I think there's a larger teachable moment here, which is that ultimately, we're allowed to make whatever definitions we want in math (as long as we clearly communicate to others what our definitions are).  In particular, you might think that the inability to say that $S_1=1/2$ is a flaw with the standard definition of infinite sums, so you should make a better one.  Maybe you don't know how to give a completely precise definition of what your better definition of infinite sums should be, but you can write down some properties it ought to have.  Maybe one of them is that if the partial sums go back and forth between two numbers, the infinite sum should be the average of them.  You can carefully go through the argument and write down a list of axioms that infinite sums need to satisfy in order for the argument to go through (among them are: a sum doesn't change when you shift its summands to the right and put a $0$ at the beginning, if you add the terms of two sums together then you get a new sum whose value is the sum of the values of the two original sums, etc.).
In the end, you get a valid proof that for any notion of "infinite sum" satisfying some reasonable-looking axioms, the sum of $1+2+3+\dots$ must be equal to $-1/12$.  At this point, you might point out another reasonable axiom for sums, which is that an infinite sum, all of whose terms are positive, must itself be positive.  If you add this axiom, you've reached a contradiction.  Thus you've proved that your seemingly reasonable list of axioms (including this last one) is self-contradictory: that is, there is no possible definition of "infinite sum" that can satisfy all your axioms.  This is good evidence that maybe some of your axioms aren't quite as reasonable as you thought, and maybe you should stick to the standard definition.  (In fact, it turns out that you don't even need to assume the last positivity axiom to run into problems.  See the discussion in ziggurism's answer, in particular this nice argument on Wikipedia that it links to.)
A: I've argued this with others unsuccessfully, but I'm a glutton for punishment.
What you are saying is 

assume that 
$$1 - 1 + 1 - 1 + 1 - 1 + \ldots$$ is a convergent infinite 
      series 
and let  $$S_1 = 1 - 1 + 1 - 1 + 1 - 1 + \ldots$$

The assumption that 
$$ 1 - 1 + 1 - 1 + 1 - 1 + \ldots$$ is a real number is a false hypothesis.  Convergent series are very rigorously defined and $1 - 1 + 1 - 1 + 1 - 1 + \ldots$ isn't one.
A: This is a very good question. Most responses to the video instruct the viewer to use zeta function regularization or Ramanujan summation, without actually engaging the proof and showing why it's wrong. Or their demonstrations why it's wrong ignore the entire body of summability methods for divergent series.
Although they are not careful in the video to discuss notions of convergence, and which steps are compatible with which classes of divergent series, some of their steps can be justified. For example, the series $1-1+1-1+\dotsb$ is Cesàro summable, which is a stable and linear summation method, which means that the computation $1-1+1-1+\dotsb = 1 - (1-1+1-1+\dotsb)$ is justified, which shows that $1-1+1-1+\dotsb = 1/2$ (that computation isn't in the Numberphile video you linked, where they just wave their hands and say "average it", but there is also an embedded link to another Numberphile video where they do show this computation).
Similarly the computation showing $1-2+3-4+\dotsb=1/4$ can be justified by appealing to Cesàro summability.
However the series $1+2+3+\dotsb$ is not Cesàro summable, nor indeed can any stable and linear summation method sum this series without giving rise to inconsistencies. See for example this wikipedia heading or this example computation by robjohn. So when they do the subtraction $(1+2+3+\dotsb) - (1-2+3-4+\dotsb) = 0 + 4 + 0 + 8 + \dotsb,$ they are invoking linearity; they assuming a linear summation method. Then when they write $0 + 4 + 0 + 8 + 0 + \dotsb = 4 + 8 + 12 + \dotsb$ they are assuming stability. This is the error. It is inconsistent to assume both properties.
Edit: As pointed out by Mathologer, in fact stability allows you to insert only finitely many zeros into a series. So this would be an error even in the presence of Cesàro summable series.
I wonder whether their proof can be repaired by dropping the use of one of the two axioms, linearity or stability. I couldn't do it, so I asked here.

Edit: Today (13 Jan 2018) Mathologer posted a new video which I liked very much. I hope it doesn't sound like tooting my own horn if I say it seems to be a lengthy recapitulation and explication of exactly all the points in my answer here. He does improve on my answer in one way: he points out that to insert infinitely many zeros into a sum, as is done in the Numberphile video, is not valid even with a stable summation method. I called out that step as a mistake, but only because it was invoked on a sum not known to admit a stable summation method. In fact it would be a mistake anyway.
A: The operations with series are not legitimate. In modern mathematics, what it means for a series to have a sum is clear enough. 
If the author would like to operate series ad hoc like that, then he or she is accountable for creating a new set of axioms to justify the operations. For, if not, then one series may admit two different sums, which makes the theory of series opinion-based.
The operation process interesting you is like saying that $\infty - \infty = 0$. People that convince themselves that this is valid forget that subtraction operation by definition does not eat $\infty$ at all. If one, for some reason, does want to write $\infty - \infty = 0$, then he or she should give a reasonable definition of subtraction.
A: The theory of
assigning a value
to a series that normally
is considered to diverge
is called the study
of "Divergent Series".
G.H.Hardy wrote a well-known book
with this title.
A simple way to assign a value
is to have a series
that converges for a limited range
of its parameter,
and see what happens when
a value is assigned
where the series does not converge.
For example:
$\sum_{n=0}^{\infty} (-1)^nx^n
=\frac1{1+x}
$.
If we set $x=1$,
we get
$1-1+1-1+1... = \frac12
$.
If we set $x=2$,
we get
$1-2+4-8+16... = \frac13
$.
There are lots more advanced topics
in this field.
Ramanujan did a bunch.
It can be fun.
