How important are design patterns in web development?
Knowing design patterns won't be much use until you know why they are the best strategy for a given problem. Learning design patterns from the very beginning is probably fine, except you've missed all the "wrong" ways to solve that problem, which in turn means you may be missing subtle difference in when to use the given pattern and when to not use it.
The only thing worse than people who stick to their old ways and don't bother learning the proper way, is people who learn the proper way and don't bother learning why that way is proper. And they keep applying it to stuff that it shouldn't be applied to, because they just don't know better.
So my point is, if you're new at web development, don't be too caught up in the design pattern hype (though it's a good hype). Learn by doing stuff yourself. When you've reached a certain level, read up on design patterns and see where they could have been applied to have made your code better.
THAT is how you learn them properly. Not like being forced to run before you can walk.
What are the design patterns that I should be completely familiar with? And what is one easy example that each can be used for?
I am a web developer (I use Django, and is familiar with separation of logic), but I work at a Desktop-app company.  They are always talking about singletons, and I forget...but it leaves me no clue!