How do I see the currently running applications on an iPod Touch?
In iOS 3, you press the home button, with very few and only Apple-engineered exceptions (eg. the iPod app) the app is closed.

In iOS 4+, there is limited multitasking available to third-party apps. Apple freezes all processes in the background unless they're performing one of the following functions (ripped from Ars):


Audio: you can now listen to streaming music from apps, like Pandora or newscasts through the NPR app, while doing other activities on your phone.
VoIP: you can carry on Voice Over IP calls on services like Skype without having to quit the app if you need to perform other tasks.
Location: apps that need to poll your location, such as GPS and direction apps, will be able to do so in the background.
Local notifications: third-party apps no longer have to rely solely on push notifications if they want to alert you of something on your phone. If you have an alert set in, say, one of Omni's applications, it can ping you when the time comes instead of having to go through a convoluted series of Internet tubes to get to you.
Complete tasks: if you start a task in an application and then switch to another one (such as downloading a new map in your favorite game), it can now complete the task in the background instead of forcing you to sit there and wait on it.
Fast app switching: this is basically "pausing" an app where it is, which allows you to quickly switch away from it and then switch back, picking up where you left off.


In iOS 4, double-tapping the home button shows a tray holding both recently-used apps and apps performing the above limited-multitasking functions. You can close them/remove them from the tray by clicking the red "-" sign.