Cloud use service model : IaaS (Infrastructure as a service) , PaaS(Platform as a service) , SaaS (Software as a service ).
Simple example that use cloud is : Microsoft Azure , Amazon's Ec2 , google drive
Cloud is only application that provide user infrastructure , platform or software online with user-defined requirement and is scalable . that is maintained and upgraded by cloud provider.
user have to pay only for what they are using.
Your Answers :
--> Is it just a website hosted by a huge datacenter? A. No , it is not website. it is service that enable user to store data , work with software online.
--> Can I only access Cloud Application through a web browser then? A. Most probably you can use through web browser.
--> Is it possible to have a desktop application 'running' in the cloud? If so, what kind of code is running in the cloud? A. Yes , the applications that run on cloud also contain desktop application. E.g , word editor or photo editor is desktop application and if you need for your cloud that then cloud provider provide you that application for you.