SELinux is based on the concept of Mandatory Access Control. Under MAC, administrators control all interactions of software on the system. The concept of least privilege is used, by default applications and users have no rights, as all rights must be granted by an administrator as part of the system's security policy.
This contrasts with the Discretionary Access Control concept that is the standard Linux security model. Under DAC, files are owned by a user and that user has full control over them. An attacker who penetrates an account can do anything with the files owned by that user. For example, an attacker penetration a web server has full control over all files owned by the webserver account. Worse, if an application runs under the context of the root user, an attacker penetrating it now has full control over the entire system.
MAC in effect provides each application with a virtual sandbox that only allows the application to perform the tasks it is designed for and explicitly allowed in the security policy to perform. For example, the webserver process may only be able to read web published files and serve them on a specified network port. An attacker penetrating it will not be able to perform any activities not expressly permitted to the process by the security policy, even if the process is running as the root user.
Standard Unix permissions are still present on the system, and will be consulted before the SELinux policy when access attempts are made. If the standard permissions would deny access, access is simply denied and SELinux is not involved. If the standard file permissions would allow access, the SELinux policy is consulted and access is either allowed or denied based on the security contexts of the source process and the targeted object.