In an active/active configuration, systems also use a backup but will manage both the main and redundant systems concurrently. This way, if there is a failure the user is unlikely to notice. The backup system is already online, and takes on increased load while the main system is fixed and brought back online.
Typically, an active/active installation for a stateless service would maintain a redundant instance, and requests are load balanced using a virtual IP address and a load balancer such as HAProxy.
A typical active/active installation for a stateful service would include redundant services with all instances having an identical state. For example, updates to one instance of a database would also update all other instances. This way a request to one instance is the same as a request to any other. A load balancer manages the traffic to these systems, ensuring that operational systems always handle the request.
These are some of the more common ways to implement these high availability architectures, but they are by no means the only ways to do it. The important thing is to make sure that your services are redundant, and available; how you achieve that is up to you. This document will cover some of the more common options for highly available systems.