SVMs and solution

In SVMs, is the solution to the minimization problem $$\textbf{w} = \sum_{i=1}^{n} \alpha_i x_i y_i$$ and once we know $\textbf{w}$ we can get $\textbf{b}$?

In plain English can somebody please describe the solution above? Basically the hyperplane that separates the data with the maximum margin is a linear combination of the training data?

The quantity $\textbf{b}$ is the distance from the decision boundary to the closest training example?

I have not been able to find a simple explanation of the solution above in the books and notes I have looked at.

$b$ can be calculated based on any support vector that lies on the boundary; in particular having $0 < \alpha_i < C$. For such a vector $x_i$, we know that $y_i (w^T x_i + b) = 1$, and the value for $b$ can simply be calculated (so yes, you do indeed need $w$ to calculate $b$). Numerically, one generally averages over the values one gets from such a calculation.
There are alternative methods for calculating; see for example Section 4.1.5 in the LibSVM paper, they handle the corner case as well where there no support vectors having $0 < \alpha_i < C$.