What does the notation $\binom{n}{i}$ mean? What do the parentheses next to the summation involving the binomial coefficients mean? Like this:
$$\sum  _{i=0}^{n} \binom{n}{i}a^{(n-i)}b^i=\left(a+b\right)^n $$
 A: The way it works if you expand $(a+b)^n$ is that there are $n$ brackets and you have to choose an $a$ or a $b$ from each one. There are 2 choices for each bracket, hence 2^n choices overall.
The sum you have given gathers all the terms with $i$ copies of $a$ and $n-i$ copies of $b$. The bracket $\binom{n}{i}$ is the number of ways of doing this, and is called n-Choose-i (which is the reason for the $C$ in some alternative notations) or a Binomial Coefficient (binomial because the original brackets contain two terms $a$ and $b$).
It might be interesting for you to see whether you can get any intuition or insight for why the value of the bracket is what it is.
A: They are not parentheses. It is the binomial coefficient, as Mark Bennet commented.  
$$\binom{n}{i}=\dfrac{n!}{i!(n-i)!}.$$
Given $n$ objects it counts the number of ways of choosing $i$ objects from those $n$, i.e. the combinations of $n$ objects taken $i$ at a time without repetitions.
Alternative notation: $C(n,i)$ or $C_i^n$ (or $C_n^i$).
A: It is read "$n$ choose $i$" and is equal to the number of ways to choose $i$ objects out of $n$ different objects, where the objects are pairwise different.  This number is given by the formula
\begin{equation}
{n \choose i}=\frac{n!}{i!(n-i)!}.
\end{equation}
