Einstein summation convention dictates that repeated indices should be summed. Thus the equation
$a_{ij} = b_{ik}c_{kj}$
is taken to mean
$a_{ij} = \sum_k b_{ik}c_{kj}$
where in both cases the range of summation is implicit.
Oftimes when I have come across this notation, it is followed by the statement "where summation over index $k$ is implicit." This seems to defeat the point of Einstein notation (to reduce clutter in equations).
Othertimes, the summation is not obvious (as it may be above). For example, if asked to evaluate $F^{\mu \nu}F_{\mu \nu}$, one might think that the answer depends on the values of $\mu$ and $\nu$, but in actual fact, summation is implied.
Given these ambiguities and failure to reduce clutter (well, rather trading clutter in equations to clutter in text), why should one use Einstein notation?
Best Answer
What is Einstein's summation notation?
While Einstein may have taken it to be simply a convention to sum "any repeated indices", as Zev Chronocles alluded to in a comment, such a summation convention would not satisfy the "makes it impossible to write down anything that is not coordinate-independent" property that proponents of the convention often claim.
In modern geometric language, one should think of Einstein's summation convention as a very precise way to express the natural duality pairings/contractions when looking at a multilinear object.
More precisely: let $V$ be some vector space and $V^*$ its dual. There is a natural bilinear operation taking $v\in V$ and $\omega\in V^*$ to obtain a scalar value $\omega(v)$; this could alternatively be denoted as $\omega\cdot v$ or $\langle \omega,v\rangle$. This duality pairing can also be called contraction and sometimes denoted by $\mathfrak{c}: V\otimes V^* \to \mathbb{R}$ (or different scalar field if your vector space is over some other field).
Now, letting $\eta$ be an arbitrary element of $V^{p,q}:= (\otimes^p V)\otimes (\otimes^q V^*)$, as long as $p,q$ are both positive, we can take a contraction between any one factor of $V$ against any other factor of $V^*$. Each one of these contractions give a mapping $V^{p,q} \to V^{p-1,q-1}$, and it is tedious to name every one of them (you can index each one by calling $\mathfrak{c}_{i,j}$ the contraction between the $i$th factor of $V$ with the $j$th factor of $V^*$).
The Einstein convention gets around this by being an index convention, where $\eta$ is written as $\eta^{i_1\cdots i_p}_{j_1\cdots j_q}$, an indexed object, each of the index corresponds to one of the $V$ or $V^*$ factors. Then instead of $\mathfrak{c}_{i,j}$, we just single out the relevant factor in the index and trace over it. For example $$ \mathfrak{c}_{1,1}(\eta)^{i_1\cdots i_{p-1}}_{j_1 \cdots j_{q-1}} = \eta^{k i_1\cdots i_{p-1}}_{k j_1 \cdots j_{q-1}} $$ where the summation symbol over $k$ is suppressed. For one single tensor the advantage of this notation is not clear, but for multiple contractions, you see the advantage
$$ \mathfrak{c}_{1,1} \mathfrak{c}_{p,q} \eta = \mathfrak{c}_{p-1,q-1} \mathfrak{c}_{1,1} \eta $$
if $\eta \in V^{p,q}$. Basically, if you have multiple contractions on one expression, you will have to keep careful track of the level of contractions to put in the correct indices in the contraction symbol; in particular the symbols are not commutative. The same expression above in Einstein notation would only be
$$ \eta^{k i_1\cdots i_{p-2} \ell}_{k j_1\cdots j_{q-2} \ell} $$
and it is immediately clear which slots are contracted together. Furthermore, it is manifest that the "formulae" obtained thus are independent of the choice of basis of $V$ and $V^*$ (with respect to which we can write down the actual components of $\eta$).
What is the correct use of Einstein's notation?
Alternatives
Einstein's summation notation is ultimately about pairings between $V$ and $V^*$, so (in spite of its likely origin) you should not think of it primarily as a notation used for decluttering computations of tensor components in local coordinates, but rather as a way to efficiently solve the problem of "which two slots are we contracting again?"
From this point of view the alternatives to Einstein's notation are "invariant notation" (don't use any index; write everything in coordinate free manner) and the "Penrose diagrammatic notation" (see e.g. https://en.wikipedia.org/wiki/Penrose_graphical_notation).