"The Semantic Web is an evolving extension of the World Wide Web in which the semantics of information and services on the web is defined, making it possible for the web to understand and satisfy the requests of people and machines to use the web content." -Wikipedia
In a nutshell, the Semantic Web is about enabling computers to process information from the World Wide Web. On the web, text documents link to other text documents. These hyperlinks are accessed through a word, phrase, or image on the document; as we all know, these links are intended for a human user to follow by clicking. We usually know what the link means by contextual clues in the document. However, if you intend for computers to process this information, the exact relationship between the two documents is not always clear. Even within a single document, words and sentences have ambiguous meaning, and computer processing of text has a long way to go before reaching the level of human understanding. When information is presented in the form of images and spatial relationships, our computers cannot make any sense of it at all.
Contrast this with relational databases where everything is structured. If our software is aware of the database schema, we can read the data with almost any programming language. The database can also be placed on the Internet without a lot of trouble. But the vast amount of information is not openly available in this form. Even if it were, we would still have the problem of millions of different schemas, each with different semantics. One thing we need is a way to say that "Car" in one database is the same as "Automobile" in another. The relationships with other concepts also need to be defined. For example, computers might need to know that a car is a vehicle, that airplanes and helicopters are also vehicles, and that vehicles can transport things. With this kind of mapping, we have a chance of being able to build complex systems that use data from different databases interchangably. Consider a logistics program that might make decisions based based on map data from many different vendors, along with their shipping capabilities.
The Semantic Web is an attempt to make data open, linked, and with a shared semantics. Data can be things that are internal to your organization, your business partners, or the whole world. It can describe your processes, your employees, your products, or anything at all. This does not mean that everyone can see it, only that we have the ability to share as necessary. Which data is shared with whom, and how, is determined by your requirements. The Semantic Web provides a framework for doing all of this.
Here are a few links to get you started in learning more about the Semantic Web.
Wikipedia article on the Semantic Web
The official W3C Semantic Web site
A short video introduction to the Semantic Web