Where do I store and download for IPhone Apps and Computer Apps?

I'm feeling like a dinosaur in not understanding how to download and store apps for my iphone and those that I may DEMO or buy for my computer.
For instance I have started syncing the iPhone to the MacBookPro so that this is easy when I am traveling. Now the MacBookPro is in the shop and I'd like to d/l an app for the iPhone. Can I do this on the MacPro which I am using now? If so, do I go directly to the Mac App Store do I open some App program on my MacPro? Do I have to d/l this app from my phone or MacBookPro only? I don't get this.
Also, I called in for Aperture support the other day and didn't have any info on this software that I bought. I guess I downloaded a demo from some Mac Computer App store and then paid for the App after running the Demo? Then this information is stored somewhere different from having downloaded the Software directly without a demo or from another site or from some app on my MacPro?
Something of a newbie here so any straightening out on how this works now and (perhaps) how it may work differently in IOS 5 or Lion would be really wonderful.

The iphone will sync with one computer at a time.  If you sync to another, then the current contetn will be erased and reaplaqced with content from the new computer.
"If so, do I go directly to the Mac App Store do I open some App program on my MacPro? "
The Mac app  store has NOTHING at all to do with iphone.  it is for buyong apps for your computer ONLY, NOT iphone.
You can download apps from the itunes app store on your iphone or on your computer.  Your choice.
iPhone User Guide (For iOS 5.0 Software)

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