Clarifying Dual Representation

$$w = (X^T X + \lambda I_n)^{-1} X^T y \rightarrow \sum^m_{i=1} \alpha_i x_i \\ \alpha = (\alpha_1, ..., \alpha_m)^T \rightarrow (XX^T + \lambda I_m)^{-1} y$$

From what I understand, the above demonstrates the conversion from primal to dual form. However, I am at a loss as to the definition and meaning of each of these forms and how the transformation takes place. I would appreciate help in understanding:

1. What is the definition of primal and dual representation?
2. How does one convert between primal and dual form?
3. What benefit does the conversion offer?