How do I turn this off? While I was in edit mode I tried to extrude but accidentally pressed W instead of E and now I don't know how to turn this thing off.

It makes my cursor select everything inside the little circle, making it very hard for me to model something with many vertices.
W cycles the mode of the 'Select' tool (the topmost icon in the toolshelf). Available modes are Select Box, Select Circle, Select Lasso, Tweak. In your screenshot you can see that it's set to 'Select Circle' as you can also see the 'Radius' setting that lets you control how large the selection region is.
Simply press W again until you get back to 'Select Box' mode.