Authentication is the process of determining if someone (or something) is what they claim to be. In the context of computing, the credentials provided by the person requesting access to a resource are compared to those held by the provider of that resource. If the credentials match, the person is given access to the resource.
Someone is authenticated based on what they know (e.g. a password), what they have (e.g. a security token) or what they are (e.g. a fingerprint). These methods may be combined to provide a greater level of security known as two-factor, or multi-factor, authentication.