By previous question, if there is a elementary embedding from $\mathfrak A$ into $\mathfrak B$, then $\mathfrak A \equiv \mathfrak B$.
Now it is naturally to ask conversely, if $\mathfrak A \equiv \mathfrak B$, is there a elementary embedding to link them? Or there are $\mathfrak A,\mathfrak B$ such that $\mathfrak A \equiv \mathfrak B$ but none can be embedded to the other.
For example, real field $\mathbb R$ and hyperreal field $\mathbb R^*$ are elementary equivalent but $\mathbb R \prec \mathbb R^*$.