I am building an application which would be good if a user
could 'activate' it over the internet (similar to how you have to
activate Adobe products before using them). Is this possible? If
so, how would I go about doing this?
There's no built-in way to do this. You'd have to write it.
(Also, AIR does not let you generate a machine specific UID without
which uniquely identifying the machine that the application is
installed on may be hard).