"Cloud" is one of those unfortunate words like "enterprise" that got popular with managers and was subsequently overused to the point of meaninglessness. 99% of the time it simply means "hosted". There is a real meaning having to do with computing power as a fungible commodity, but as I'll explain people rarely mean that.
AWS and salesforce are cloud services that you pay to host your cloud systems. Azure is a technology by Microsoft that helps vendors like AWS and salesforce make their product.
For you, a cloud system is just like any other, except that it's not physically in your server room and you'll probably amortize the costs differently.
You just develop the Web app. The actually meaningful sense of "cloud" had to do with the ability to scale up and down on demand, which is not something you want to worry about much at the beginning, except in that you want to make your app scalable (but you always want to do that anyways).
The idea behind cloud was that if your one application server starts getting overloaded, you spin up another one to take some pressure off of it, and spin it back down when the traffic goes back down. Great idea; very useful for some people, but then 99% of the time, people mean "a remote virtual host".