Web 2.0 refers to a general shift in the perception and use of Internet-based tools. Although the name is reminiscent of the version numbers that denote various software updates, Web 2.0 does not specifically refer to the upgrade of any one technology. Generally, "Web 1.0" websites were designed with passive viewing in mind. Web 2.0 websties, on the other hand, tend to encourage interaction and collaboration through user-generated content. The idea is to shift from passively absorbing available media to actively contributing and customizing said media. Typical examples of Web 2.0 include popular social networking sites, blogs and wikis.
The term "Web 2.0" began to gain popularity after O'Reilly Media hosted the Web 2.0 Conference in 2004. Web 2.0 moves away from the conventional notion of centralized websites and focuses on shifting "power" to desktops/mobile devices. One commonly cited example is Encyclopedia Britannica vs Wikipedia. Encyclopedia Britannica represents the old paradigm, where only experts are allowed to author articles which are released to the consumers in subsequent updates. Wikipedia, on the other hand, relies on the contributions of everyday users.
Examples of Web 2.0 tools include RSS feeds and tagging. RSS feeds allow users to recieve updates from websites. Moreover, these updates can be narrowed by keywords or content to specifically tailor to the user. Tagging helps sort and classify information, which helps develop communities of like-minded users.
The image is a comparison taken from O'Reilly's article on Web 2.0.