How important are design patterns in web development?
For web applications, understanding at least at a rudimentary level the patterns described in Patterns of Enterprise Application Architecture has proven valuable to me. Gang of Four patterns are worth knowing, too.
But I would argue that you simply don't need encyclopedic knowledge of patterns to get started. A cursory understanding will help you understand where to look when you start to encounter friction between your ideas/business problems and your code. I had a couple of weekend trips that allowed me to plow through these two books in their entirety, but I still find the detailed information in the patterns section more useful as a reference than as background knowledge.
Reading just the "Part 1" sections of the GoF or PoEAA will help you far more than learning three or four patterns in depth, because you'll know where to look when you encounter problems they describe. And you can look up the details of most of the patterns they describe online.
GoF patterns that I use directly or indirectly, often unconsciously, in web development, include: Observer, Command, Composite, State, Strategy. I usually don't use Singleton except as a client of logging and service locator/dependency injection tools. PoEAA patterns that I use regularly, usually unconsciously, or incidentally as a part of the data access strategy or web framework I'm using, are Active Record, Application Controller, Data Mapper, Domain Model, Gateway, Lazy Load, Layer Supertype, Page Controller, Template View, and Value Object. That's not exhaustive; these are just a few that popped into mind.
Most of those are probably more usefully learned by starting with an opinionated web development framework, like Rails, Django, or Castle Monorail, than in the abstract. After all, patterns were identified and extracted from thousands of successful app development experiences, not invented and then glued on because they sounded clever.
It's pretty easy to get overly excited by better-than-superficial knowledge gained on one or two patterns and then seeing "only nails" for every problem you see shortly thereafter and trying to hammer an ill-fitting pattern into a solution because you understand how it works.
So, learn patterns, yes; get a superficial overview of the motivations of all of the commonly used ones, but don't feel like you have to wait to write serious code until you understand some arbitrary list of them.
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!