I don't know what 1+2+3+4+...=-1/12 has to do with p-adic numbers. Care to explain?

Because using the "normal"/simple definitions of summation and integers, that summation does diverge. The natural numbers are closed over addition, and yet -1/12 is not a natural number, which breaks that closure of natural numbers. So in order to make sense of this contradiction, alternative/fancier definitions of summations and numbers must be used. Specifically, p-adic numbers, which converge for large values rather than diverge. Once you're using p-adic numbers, you're not using the natural numbers and aren't restricted to the closure of natural numbers, and so can achieve 1 + 2 + 3 + ... = -1/12.

Isn't 2-1 equivalent to 1/2, so 1/2-1 would be 1/(1/2), which would be equivalent to 2? So 1/1-1 + 1/2-1 + 1/3-1 + 1/4-1 ... = 1+2+3+4...?

Like I said, I'm not good at the math, so it's interesting why two seemingly equivalent statements aren't equivalent.
i.e. sum n where n=1..? != sum 1/(1/n) where n=1..?

As others have said, you can't treat infinity like a variable and do algebra with it. You can do some things (which actually involve evaluating a limit), but there are several things one might be tempted to do with infinity that would seem valid, but in reality aren't.

But maybe I'm not understanding, as I can't see the contradiction/inequality in sum n where n=1..? != sum 1/(1/n) where n=1..?.

That question of mine was in response to Mats1's post above it, where he told me that one of those sums did equal -1/12, but the other did not. Seeing they're equivalent, I was asking why this is so.

As for treating infinity as a regular algebraic variable - yeah I've seen that used to "prove" that 0=1, etc...

The answer is that you are correct: Those two things are the same, and however you define one of them, you should get the same answer as for the other.

I thought however, that when presented with the naive sum 1 + 2 + 3 + 4... This will diverge. However, a value can be assigned to this series, after some manipulation, that is - 1/12, but it does not reflect the simple summation of the terms of the sequence. I'm no expert of the zeta function, but I was pretty sure this is how it goes down.

@Mats1: As I said ealier, the result has been obtained using more complicated methods based on complex analysis and not using some kind of series manipulation. When you sum an infinite number of terms you have to be very careful to what you do. Most manipulations gives in fact wrong results.

A number must fullfill some rules to be considered a number. Eg zero is the unit value of the additive operation, like 1 is the unit value of multiplication operation. Therefor

Addition:
(1) x + 0 = x = 0 +x
and
(2) x + (-x) = 0 = (-x) + x 

Now lets check infinit

Addition:
(1) inf + 0 = inf = 0 + inf

But what about this ?
(2) inf + (-inf) = 0 

If for a infinit number inf = inf + X (eg inf = 1+inf) would hold true and inf would be a valid number, then following should work
      inf + (-inf) 
= 1 + inf + (-inf) 
= 1 +     0 
= 1 
!= 0 
= inf + (-inf)

Even simpe add operations would not work in a mathematically way with such a definition of infinit.

Check out surreal numbers, which have been mentioned in this thread before. They contain many infinite numbers, but they satisfy all the properties you mention.

It is weird no one has linked to this interesting video yet: