What does it mean to "prove $1+1=2$"? It is a famous bit of trivia that it took Russell and Whitehead about 300 pages to prove that $1+1 = 2$. However, this seems more like a definition rather than theorem. As far as I know, $2$ is just the symbol we use as shorthand for $1+1$, where $1$ is the multiplicative identity. 
 A: You have to define what system you are working in.  In Peano Arithmetic, PA, $1$ and $2$ are not part of the language.  They are abbreviations for $S(0)$ and $S(S(0))$, where $S$ is intended as the successor function, so you are being asked to prove $S(0)+S(0)=S(S(0))$.  You can follow the Wikipedia proof, which you may need to update a bit depending on how your version of the axioms is written.  Basically this should look a lot like your axioms that define addition.  PA does not specify a multiplicative identity, you have to prove that $S(0)$ is one by induction.
A: In the kind of type theory that Russell and Whitehead were working on, the natural definition of a cardinal number is a maximal class of equipollent sets: $1$ is the class of all sets with $1$ element, $2$ is the class of all sets with $2$ elements etc. For finite $n$, we can write down a formula $\phi_n(A)$ which holds iff $A$ has exactly $n$ elements, so that $n = \{ A \mid \phi_n(A)\}$. $1 + 1 = 2$ then becomes a meaningful "problem to prove".
A: Long comment
Without comment, the issue is not so trivial as it seems ... 
R&W's system, developed into the Principia, was aimed at the foundations of mathematics. 
The first volume is devoted to the development of mathematical logic and the basic part of a sort of "class theory" : at that time, axiomatic set theory was at its very beginning. 
Then the work goes on with the definition, on the base of logic alone and "class theory", of the arithmetical concepts : number, zero, successor, sum, etc. 
At that point, well into second volume, it was introduced the "canonical" definition [see Rob's answer above] of : $1$ as the successor of $0$ and of $2$ as the successor of $1$, and thus :

$1$ is the successor of the successor of $0$.

Then follows the proof of the fact that :

$2=1+1$,

i.e., in un-abbreviated form :


the successor of the succesor of $0$ is equal to the sum of the successor of $0$ with the successor of $0$.



With first order-arithmetic, base on Peano axioms, the basic concepts (successor, zero, sum, product) are primitive, i.e. "implicitly defined" by the axioms, but then the approach is similar : $2$ is defined as $S(S(0))$ and thus we can (and we have to) prove that :

$2=1+1$.



The meaning is :

in an axiomatized theory, having assumed the axioms as well as the definition, all other "known facts" must be proved. 

A: Here is a proof in Fitch, starting with the relevant axioms from PA:

A: Axioms for the natural numbers usually define only a single number: either $0$ or $1$. The existence of any other natural number must be inferred from these axioms using the axioms and rules of logic or set theory (if applicable). If addition on $N$ is given in your axioms, it would be a trivial exercise to prove that 1+1=2. 
Suppose, for example, you are given the axioms: 


*

*$1\in N$

*$\forall x \in N: \exists y\in N: y=x+1$


Applying both of these axioms, we can obtain:
$\exists y\in N: y=1+1$
Applying the rule of existential specification (for $y = 2$), we could infer that $2 \in N$ and $2=1+1$. This inference can be thought of as the "definition" of $2$. 
If addition on $N$ is not given in your axioms, you would have to construct (i.e. prove the existence of such a function) using the axioms of your set theory before you could "define" or infer that $2=1+1$ as above.
