It depends on what profile information you need to display.
Basically anything in a user's profile that's public-facing is exposed via the API and requires no action on behalf of the end-user. Starting with v2.0 of the API, you have limited access to some private user information (such as the their full reputation history) through the OAuth 2.0 methods.
In either case, you will need to register your application herehere. This will provide you with an API key (essentially required to use the public API) and a client ID/secret (required for the OAuth 2.0 methods).
If you choose to use the OAuth 2.0 methods, your users will be prompted to grant certain permissions to your application, such as read-only access to their private profile information or access to their inbox. The prompt looks something like this: