Optimization is such an important subject in optical design that we need to say more about it, even though it was briefly described under How to Design a Lens. Remember that the goal of optimization is to take a starting lens of some sort and change it to improve its performance (the starting lens should have a suitable number of optical surfaces of suitable types, since optimization can change only the values of the parameters, not the number or types of surfaces). Since optics is very precise (distances of micrometers can make a big difference), we need to closely determine the values of all our variables at each step of the optimization.
Let's consider local optimization first. What does "local" mean? If you have a lens model, an error function is something that correlates with its image performance, like spot size or RMS wavefront error -- smaller is better. As variables are changed, the lens changes, ray trace values change, and the error function takes on new values. If you could plot these out, you would create a map of the hills and valleys of error function space (in anywhere from one to 99 dimensions or more, depending on your variables). In the admittedly silly sketch above, vertical distance represents the error function value (lower is better), and horizontal position represents ONE of the variables in the lens (for example, it could be the curvature of the front surface).
Since smaller is better, your goal is to find the lowest possible point on this map -- the Death Valley of Error Function Land (EFL). Local optimization finds the lowest near-by region in the EFL, so if you are lucky (or smart) in choosing your starting point, you will do well (by analogy, starting in Los Angeles might let you reach Death Valley using local optimization, but starting in New York would not -- you'd probably end up somewhere in New Jersey). Does this analogy help? Maybe not, but the point is, with local optimization, your choice of starting point is very important. (In our picture, local optimization will NOT get you to the lowest point -- it will roll you into one of the valleys to the right or left of the "You are here" starting point).
Now consider global optimization. This is an algorithm that somehow looks at the entire map of Error Function Land and (eventually) locates the lowest point regardless of where you started. Even if you start in Florida, global optimization will eventually get you to Death Valley, though depending on the methods used, it might take a really long time to actually get there, and you might be told about a lot of other low places along the way, some of which might be low enough for your purposes. Silly analogy? Maybe, but the point to remember here is that global optimization considers the whole of "error function space," so your actual starting point is much less critical. (In our picture, global optimization should take you to the desired low point).