The best answer is "it depends", or perhaps "it's complicated".
Originally Posted by kingzogofalbania
By default, AIR and WebWorks apps will basically stop receiving any events while the tablet is in standby. They would act much as they would when not active (in either Paused or Default modes, as opposed to Showcase mode).
I'm not sure about native apps, but my guess is they are unaffected by this unless they explicitly listen for and respond to a "standby" event by putting themselves into an "inactive" state (if that's appropriate). If that's true, it would be a "best practice" for most native apps to do this to avoid wasting battery in standby. (The new games from EA don't follow this guidance, and actually suck an unacceptable amount of power when inactive, including in standby. Folks should ask them to fix that.)
The browser you run is an AIR app wrapped around the native "webview" component, so what happens in that component is probably pretty flexible. In recent versions of the OS, streaming audio will continue to run, though I believe video halts. Generally any network activity other than audio is probably halted, though this may be under the control of the browser in some way I'm not aware of.
In a more general sense, the answer is basically that there's no single answer that applies to everything. It's not like the OS physically freezes everything... developers can definitely take advantage of various options to keep running.
My own apps do this in several ways. White Noise continues generating sound whether the tablet is awake or not, so it's not directly affected by Standby other than to halt any UI activity that would be a waste. Battery Guru is a bit unusual in that it is almost always acting as though the tablet were in Standby, to ensure it has a negligible effect on battery consumption. When the tablet is really in Standby, it does nothing except wake up every two minutes for a few milliseconds to record the new battery/power readings to its database.
Other apps can also request that the OS keep certain services active in Standby, so that they can continue to work. This currently includes GPS, networking, and the accelerometer, and some others are undocumented or (I'm guessing) in the works. If no app has requested a given resource, the OS will "park" it while in Standby, so that it consumes no or few resources. For example, if nothing is actively using WiFi, then after a while in Standby the WiFi is effectively shut off or put in a very reduced (probably receive-only) mode that uses little power.