What Constitutes a Compute Hour or Cost/Hour

Good answers by David and Mike. Let me add a bit: Rather than thinking in terms of apps, think about virtual machines. When that virtual machine is spun up (even in a stopped state, the virtual machine is allocated), you're being charged. You can easily start and stop apps within that virtual machine instance: Maybe it's a web application, background rendering process, whatever. Whether you're burning 0% CPU or 100% CPU, you're paying in clock-hours for that virtual machine.

Note that even with Cloud Services (web and worker roles), you're ultimately dealing with virtual machines. Each instance of a web role or worker role is its own Windows Server virtual machine.

Oh, and same goes for Web Sites reserved instances.

Regarding your question of switching an application off to save billable hours: If, say, you have a nightly process that runs for maybe an hour or two, then consider placing that process in its own Cloud Service (or Virtual Machine). When the time comes to run it, deploy it to its named slot (yourname.cloudapp.net), let it run until the task is done, then delete the deployment. Now the virtual machine instances hosting your process runs for 2 hours, not 24. What you can not do is reduce the count of a web or worker role to zero. So if you have, say, a web role with 2 instances and a worker role to do some nightly stuff, you cannot set the count to zero worker role instances upon completion of the task. That's why I suggested placing the nightly task in its own deployment.

EDIT Things have changed a bit since over the past few months. In particular, the following was announced in June 2013:

• Billing is now per-minute instead of per-hour. This applies to Virtual Machines, Cloud Services (web/worker), Web Sites, and Mobile Services.
• You may now stop Virtual Machines without deleting them, and billing stops while the VM is stopped. Note: If no other Virtual Machines are running in the same deployment, you'll lose your assigned IP address. To prevent IP-loss, you would need to stop the Virtual Machine but leave it in a provisioned state, which means you'd still be billed for the stopped VM but you'd preserve your IP address. Again, this caveat only applies to scenarios where there's no other Virtual Machines running.

You can read ScottGu's blog post about the per-minute and stopped-VM updates here.

An hour counts as the time your app is deployed (running or not, being accessed or not). An hour is a 'wall clock' hour - meaning if you deploy at 10:45am and deleted at 11:05am, you are charged two hours (the 10 o'clock hour and the 11 o'clock hour).

To be more clear, the time your app is deployed means the time the server/s instance/s that host your app is active. Effectively the time is measured in clock hours.

[Edited] A "stopped" VM on Azure Virtual Machines (IaaS model) no longer consumes compute hours, you will only be billed by the storage/managed disks until you decide to delete the VM.