What Is a Process?
A process is an activity that can be performed by 1 or more applications. When you think of an application actually doing something, for example playing music, posting a message to facebook or syncing your rss feed, these are processes. However, it is important to note that just because a process exists, does not mean it may be actually; doing anything. If you like, it may be easier to say that processes may be in an idle or active state.
What Is an Application?
An application is something that makes use of many different processes to provide you with some wanted functionality. For example, the official facebook or twitter apps. An application may also be idle or active, depending if it has any active processes currently associated with it.
When you leave an application, that application is allowed to keep its process running in the background (true multitasking) allowing it to carry on doing whatever work it needs to do. For example continue to download the rest of a webpage or playing music. However, just because an application leaves processes ?running? in the background, does not mean they are actually doing anything at all. They are being kept in memory just in case you are going to use them again soon. Many people cite this as a battery drain, nothing could be further from the truth however. Storing a footprint of an application in memory uses exactly the same amount of battery as it would if that section of memory is free. If you continue to open applications, then more of your memory will be used.
Eventually there will be no memory left, time to use a task killer? No! Android is smart enough to recognise when it is running low on available memory, and will start to close those apps that it deems are low priority. The way it determines priority means that those apps you have used least, and are not core to the phone will be closed down first. This does not include apps that are currently in the foreground, or as mentioned core apps such as the clock alarm. When android does close apps itself to free up memory, it does this in a very clever way in that the next time a closed app is reopened, it will restore it as if it had never been closed in the first place (this is similar to what iOS actually calls it?;s main multitasking, laughable I know).
The key point to take from the above is, that task killers are completely not required, and actually interfere with how the android os works under the hood.