With tags like <b> or <font>, the meaning of the tags themselves as they're supposed to be understood by the browser is clear. Semantic tags, on the other hand, are supposed to associate meaning to their content.

On the web, though, "semantics" is applied to the problem domain; i.e. "what application am I trying to build with this code", i.e. "what is the meaning of this word with respect to the final user-facing application". Thus, for HTML you get words for the things that are typically build with HTML: web pages. Then you get tags for articles, images, quotes, headers and footers, side bars, etc. These describe the types of content in the page, not how they look.

The "semantic web" is a generalization of this "domain knowledge" idea, providing tools to define your own vocabularies and build applications with them. This approach amounts to creating your own programming language (a new one for each application) in a way that best suit the problem at hand, rather than using a general purpose programming language.