While the basic idea behind multitasking is the same on your smartphone as it is on your PC, how it works--both under the hood and in terms of the interface--can be drastically different.
When it comes to mobile operating systems, "multitasking" is used to refer to two separate features: The ability to run applications in the "background" (that is, apps running behind the current app, as is the case on a PC), and the ability to save the state of apps to disk (effectively pausing them).
Generally speaking, apps that continue to run in the background require more system resources (memory, processing power, etc) than an app that has been "paused," so managing these behaviors can be critical to keeping your smartphone responsive, and preventing its battery from draining too quickly.
A simple example might be a music-playing application; you'll probably expect it to continually run in the background, playing music, regardless of what other apps you've opened since you started playing music. On the other hand, a calendar application may only need to be updated every few minutes to check for upcoming appointments. In between periodic updates, the app may be paused rather than continue to actively consume resources.