How important are design patterns in web development?
Honestly, patterns are important but knowing when to use them is just as important.  There is never going to be any set answer it is something you need to feel out for yourself.  People that fight about  it being an absolute where you should always use them or always not use them are incorrect.  Design patterns are a tool. I would suggest looking at Amazon.com for a book in whatever language you are writing in that deals specifically with design patterns. I know there is one written for Ruby on Rails that is great though I don't remember the name, there is also one for Java called Head First Design Patterns, and on for C# written by Bob and Micah Martin that is excellent.  Read whichever one of those that applies to the language you are most familiar with.  Even if you don't use all of the patterns it is good to understand how they work and when they will be useful to use.
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!