The expression
\begin{align}
\alpha|1\rangle+\beta|0\rangle
\end{align}
is the state of a single qubit written as a linear combination of the state $|1\rangle$ and the state $|0\rangle$. If you were to make a measurement on this qubit, then you would either return $1$ or $0$ with probabilities $|\alpha|^2$ and $|\beta|^2$ respectively.

The expression
\begin{align}
(\alpha|1\rangle + \beta|0\rangle)^N
\end{align}
is probably a shorthand for
\begin{align}
\underbrace{(\alpha|1\rangle + \beta|1\rangle)\otimes\cdots \otimes(\alpha|1\rangle + \beta|0\rangle)}_{N\,\text{factors}},{}{}
\end{align}
namely the $N$-fold tensor product of the state $\alpha|1\rangle + \beta|1\rangle$ with itself. This represents the state of $N$ qubits.

If you make a measurement on a system of $N$ such qubits, then you will obtain one of $2^N$ possibilities, namely the $2^N$ distinct sequences of $1$'s and $0$'s obtained by expanding out the product. The probability of obtaining such a sequence is its associated coefficient. In fact, in this case, the probability of measuring a single such sequence is $|\alpha|^n|\beta|^{N-n}$ where $n$ is the number of $1$'s in the sequence, and therefore $N-n$ is the number of $0$'s in the sequence. So, for example, the state
\begin{align}
|1\rangle|1\rangle\underbrace{|0\rangle\cdots |0\rangle}_{N-2\,\text{factors}}
\end{align}
has associated probability $|\alpha|^2|\beta|^{N-2}$ of measurement.

## Best Answer

It is basically a measure of the quantumness of some correlations, which is not vanishing for some separable state. It was introduced by Ollivier and Zurek (PRL/arXiv). It is the difference between two different generalizations of the classical (Shannon) conditional entropy to the quantum world, and is 0 for a pure bipartite separable state. It has been proven to be the amount of entanglement needed in the task of state-merging (PRA/arXiv and PRA/arXiv).

Definition(PRL/arXiv) Classically the conditional entropy $H(A|B)$ is a measure of the uncertainty one has on the variable $A$ once we know the variable $B$. Of course, the definition of "knowing" $B$ becomes problematic when $B$ is quantum.

Classically, one can define $H(A|B)$ as the average $H(A|B)=\sum_b {\mathcal P}(B=b)H(A|B=b)$, each $H(A|B=b)$ being the entropy of $A$ given that the random variable has the value $b$. If one generalizes this to the quantum world, the $B=b$ part implies a quantum measurement (a POVM) which should be specified. A natural choice is the "best" measurement, the one which minimizes the entropy. The Shannon $H$ entropy is replaced by the Von Neumann entropy, and we define $S(A|B_c)=\min_{\text{POVM}} \sum_{b}\mathcal{P}(\text{POVM applied to B gives } b) S(A|\text{POVM applied to B gives }b)$.

The previous definition leads classically to a redefinition of the conditional entropy as an entropy difference : $H(A|B)=H(A,B)-H(B)$, which is always positive. Its quantum version, $S(A|B)=S(AB)-S(B)$ can be negative (in contrast with $S(A|B_c)$). Its negativity is a sufficient condition for entanglement.

The discord is defined as $S(A|B_v)-S(A|B)$ and is always positive. You can maybe see it as the amount of correlation between $A$ and $B$ which is destroyed by a classical measurement of $B$.

Link with state merging(PRA/arXiv and PRA/arXiv)

The state merging primitive is the following. Suppose Alice, Bob and Charly share a 3-party pure entangled state. Alice want to send her part to Bob without destroying the quantum correlations between $AB$ and $C$. Basically, she has to teleport $A$ to Bob, and the minimal amount of entanglement Alice and Bob need to perform this task is given by the quantum discord.