App Engine is a service provided by Google, which serves as hosting for web applications. As Google are experts in the field of scalable web applications, they provide an environment and the technology required to create applications that can scale to a large number of users. The service is currently in preview stages but it still provides a very stable working environment. With a free account there are certain quotas (database storage, bandwidth, etc…) that your application must adhere to. Specifying a budget and how it should be shared out will allow you to raise these limits if demand gets high. But you’ll find that the free quotas are very generous.
Google’s page about billing
App Engine currently supports two programming languages with which you can develop your applications. These are Python and Java but they have their limitations. Google have stripped certain parts of the languages mostly involving the ability to write the filesystem on the servers and computationally intensive functions. This allows them to ensure that your application doesn’t steal resources of other users. On the other hand the API included within App Engine allows you to access some very useful functionalities. There is the Datastore, which is similar to the database that Google uses for most of their applications, which in turn shares similarities to SQL. They provide a Memcache that allows you to store the results to popular/expensive queries in memory to improve page load speeds. These provisions are available in both Java and Python with the documentation for the latter here
Google provides excellent documentation detailing all their functionality and even have a getting started tutorial that takes you through creating a simple Guestbook application.
If you haven’t had any experience with either programming language before, try Python first as it is more forgiving towards beginners. Both languages provide the same functionality on Google App Engine, it is just a matter of preference. Finally, if you get stuck, Google’s forums should help out with any problem you may have.