Explanation
Regularization is a very important technique in machine learning to prevent overfitting. Mathematically speaking, it adds a regularization term in order to prevent the coefficients to fit so perfectly to overfit. The difference between the L1 and L2 is just that L2 is the sum of the square of the weights, while L1 is just the sum of the weights.
Much of optimization theory has historically focused on convex loss functions because they're much easier to optimize than non-convex functions: a convex function over a bounded domain is guaranteed to have a minimum, and it's easy to find that minimum by following the gradient of the function at each point no matter where you start. For non-convex functions, on the other hand, where you start matters a great deal; if you start in a bad position and follow the gradient, you're likely to end up in a local minimum that is not necessarily equal to the global minimum.
You can think of convex functions as cereal bowls: anywhere you start in the cereal bowl, you're likely to roll down to the bottom. A non-convex function is more like a skate park: lots of ramps, dips, ups and downs. It's a lot harder to find the lowest point in a skate park than it is a cereal bowl.