Web 2.0 is to the internet as post-modernism is to literature. It's the first real 'movement' of the internet and has been around now for a few years (Wikipedia suggests around 2004 - I'd dispute that).
Anyway, it's compromised of two main areas, if you will. The first being design.
Web 2.0 design is considered to be website designs which are more graphically intensive than pre 2004/5, feature more moving elements like videos, and often aim to add a sense of 3D to a design. Unfortunately, the label has become something of a stereotype and refers to glossy, embossed buttons, reflections, gradients, anything glossy with a bit of 3D-looking styling on it really. This has lead to websites trying to be very 'Web 2.0' and overdoing a lot of gloss, hence why Elliot Jay Stocks suggested we should leave it behind at a design conference.
The other aspect, more important aspect, is the interaction side. Web 2.0 websites focus on 'user-generated content' and interaction. The perfect example of this is Wikipedia, being focused around user created content. Social Networks (such as the obvious) also fall right into this category, and so do blogs and sites like this one with user submissions and commenting. Previous to 'Web 2.0' websites were static and just showed their information in a similar sense to a book.
Hope that gives you more of an idea of Web 2.0, it's on my list to publish a hub about too!