I feel a matrix moment coming on... "what is the cloud? The cloud is all around us, a prison for your program..." (what? the FAQ said bring your sense of humour...)
Ok so seriously, what is the cloud? It depends on the implementation but usual features include scalable computing resource and a charge per cpu-hour, storage area etc. So yes, it is a bit like developing on your VPS/a normal server.
As I understand it, Google App Engine allows you to consume as much as you want. The back-end resource management is done by Google and billed to you and you pay for what you use. I believe there's even a free threshold.
Amazon EC2 exposes an API that actually allows you to add virtual machine instances (someone correct me please if I'm wrong) having pre-configured them, deploy another instance of your web app, talk between private IP ranges if you wish (slicehost definitely allow this). As such, EC2 can allow you to act like a giant load balancer on the front-end passing work off to a whole number of VMs on the back end, or expose all that publicly, take your pick. I'm not sure on the exact detail because I didn't build the system but that's how I understand it.
I have a feeling (but I know least about Azure) that on Azure, resource management is done automatically, for you, by Microsoft, based on what your app uses.
So, in summary, the cloud is different things depending on which particular cloud you choose. EC2 seems to expose an API for managing resource, GAE and Azure appear to be environments which grow and shrink in the background based on your use.
Note: I am aware there are certain constraints developing in GAE, particularly with Java. In a minute, I'll edit in another thread where someone made an excellent comment on one of my posts to this effect.
Edit as promised, see this thread: http://stackoverflow.com/questions/2457735/cloud-agnostic-architecture
As for a choice of framework, it really doesn't matter as far as I'm concerned. If you are planning on deploying to one of these platforms you might want to check framework/language availability. I personally have just started Django and love it, having learnt python a while ago, so, in my totally unbiased opinion, use Django. Other developers will probably recommend other things, based on their preferences. What do you know? What are you most comfortable with? What do you like the most? I'd go with that. I chose Django purely because I'm not such a big fan of PHP, I like Python and I was comfortable with the framework when I initially played around with it.
Edit: So how do you write cloud-aware code? You design your software in such a way it fits on one of these architectures. Again, see the cloud-agnostic thread for some really good discussion on ways of doing this. For example, you might talk to some services on GAE which scale. That they are on GAE (example) doesn't really matter, you use loose coupling ideas. In essence, this is just a step up from the web service idea.
Also, another feature of the cloud I forgot to mention is the idea of CDN's being provided for you - some cloud implementations might move your data around the globe to make it more efficient to serve, or just because that's where they've got space. If that's an issue, don't use the cloud.