Explaining Applicative functor in categorical terms - monoidal functors
There are actually two uses of the word "strength" in play here. 


A strong endofunctor $F : C \to C$ over a monoidal category $(C, \otimes, I)$ is one which comes with a natural transformation $\sigma : A \otimes F(B) \to F(A \otimes B)$, satisfying some coherence conditions with respect to the associator which I will gloss over. This condition is sometimes also pronounced "$F$ has a strength".
A lax monoidal functor $F : C \to D$ is a functor between two monoidal categories $(C, \otimes, I)$ and $(D, \oplus, J)$ with natural transformations $\phi : F(A) \oplus F(B) \to F(A \otimes B)$ and $i : J \to F(I)$, again satisfying a coherence condition with respect to the associators. 
A strong monoidal functor $F : C \to D$ is one in which $\phi$ and $i$ are natural isomorphisms. That is, $F(A \otimes B) \simeq F(A) \oplus F(B)$, with $\phi$ and its inverse describing the isomorphism. 


An applicative functor, in the sense of Haskell programs, is a lax monoidal endofunctor with a strength, with the monoidal structure in question being the Cartesian products. So this is why you get the paradoxical-sounding term "strong lax monoidal functor". 

As an aside, in a Cartesian closed category, $F$ having a strength is equivalent to the existence of a natural transformation $\mathrm{map} : (A \Rightarrow B) \to (F(A) \Rightarrow F(B))$. That is, having a strength means that the functorial action is definable as a higher-order function in the programming language. 

Finally, if you're interested in the type theory of Haskell-style applicative functors, I've just blogged about it.