When using an application on iPhone and you want to exit, does pressing the home button actually fully exit the program or is it still running in the background?

I believe you have to hold down Home for several seconds to close an app.
Although I can’t imagine that all those apps you start are always “running”. My guess would be they are just in some suspended state. With the exception of some like the clock and the iPod. Which keeps ticking and playing music.

Applications completely quit when you press the home button. Their current state (what they are doing) is written to the flash memory so that it can be reloaded when the application starts again. This process is known as suspending, but the application is completely unloaded from the system memory and processor queue, so it won’t be slowing down the phone or other processes.

The limited resources on the iPhone mean that only daemons (background tasks, like the time, alarms and reminders, SMSs or push; which gets your email and other updates) stay running 24/7. These daemons then report back to the general applications such as “Clock” or “Mail” when they launch, to tell them what’s new.

Holding down the home button force quits an application (kills it without giving it any warning) and should only be used if the app stops responding.