What other axioms in set theory are stronger than AC ? I mean what are those axioms that will imply AC ?
The axiom "every set is constructible" (denoted V = L), and the axiom "every set is definable from an ordinal parameter" (denoted often as V= HOD, and sometimes as V= OD) each implies AC, and each is provably stronger than AC.
Historical Note: The axiom V = L was introduced by Gödel in his seminal work on the consistency of AC and GCH with ZF. The axiom V= HOD was first publicly introduced in a joint paper of Myhill and Scott, but their paper acknowledges that the axiom was independently considered by a number of people, including Gödel, Post, and the joint work of Vopěnka and Balcar. 


Generally speaking, let $\varphi$ be a proposition which is consistent with ZFC but not provable from ZFC, then AC+$\varphi$ is an axiom stronger than the axiom of choice. In the fashion above, we can think about $\varphi$ being "the axiom of choice holds, and $\lozenge_\kappa$ is true for all $\kappa<\aleph_\omega$." or something like that. 


Many large cardinal hypotheses imply AC, when stated in a certain natural way. For example, the assertion over ZF that there are unboundedly many inaccessible cardinals, defined as uncountable regular strong limit cardinals, implies the axiom of choice. Indeed, even having a proper class of strong limit cardinals implies the axiom of choice. What's more, the axiom of choice is equivalent over ZF to the assertion "there are unboundedly many strong limit cardinals". The forward direction is simply the usual proof in ZFC that there is a closed unbounded class of strong limit cardinals. For the reverse implication, we define that $\kappa$ is a strong limit if and only if it is an initial ordinal for which $\beta\lt\kappa\implies P(\beta)\lt\kappa$. Note that this implies in particular that the power set $P(\beta)$ is wellorderable for $\beta\lt\kappa$, since it injects into $\kappa$, which is wellorderable. Thus, if there are a proper class of such $\kappa$, then every wellorderable set will have a wellorderable power set. It is a nonobvious fact (and in fact a quite slippery fact, which beginners often get wrong when first trying to prove it) that if every wellorderable set has a wellorderable power set, then AC holds. To see that there are several natural but inequivalent formulations of inaccessibility and of strong limit cardinals in ZF, see the treatment in A. Blass, I. Dimitriou, B. Loewe, Inaccessible cardinals without the axiom of choice. 


The Axiom of Constructibility implies the Axiom of Choice, as well as many other results independent of ZFC, such as the continuum hypothesis. 


The axiom of global choice. Technically this isn't really an axiom: global choice (GC) states that there is a formula $\phi(x, y)$ such that the relation $$ A\le_\phi B:= V\models \phi(A, B)$$ is a wellordering of the universe $V$. This can't be stated as a single formula, so in some sense it's a metaaxiom. It clearly implies choice, and is implied by $V=L$: we already have a partition of the universe into $L_0$, $L_1$, . . . , $L_\alpha$, . . ., and we can get from here to a full (class)wellordering of the universe by fixing at the outset some wellordering of formulas in the language of set theory (since at each stage in the construction of $L$ we are only taking definable powersets; this is why this argument doesn't work in just $ZFC$). A couple comments on why I think global choice is interesting (even though it's not expressible in the language of set theory):


