Why Is the Result of the Convolution of a Row and a Column a Two Dimensional Matrix?
Following the wiki page on the subject, you are looking to create a 2 dimensional filter from two one-dimensional filters. 

I found that the second example was more relevant for your question. I want to give a $\begin{bmatrix}1&amp;2 &amp;1 \end{bmatrix}$ weight column wise and on top of that give a $\begin{bmatrix}1\\2 \\1 \end{bmatrix}$ weight to each row. The result is the filter  $\begin{bmatrix}1&amp;2 &amp;1\\2&amp;4&amp;2 \\1&amp;2 &amp;1 \end{bmatrix}$ which take into consideration both your demands. By the way this spesific filter will smooth you image. 

The simple fact that  this filter can be expressed as the outer product of two vectors renders this filter separable, Hence separable convolution. This post considers, also, non-separable filters.