It depends on the kind of accounts you want to log onto.
For instance there is a Google has an specific API for that Google Accounts API, other services provide similar API's some other do not.
So it depends pretty much what are your "online accounts" all about and if they do provide a public API or not.
EDIT
As per your comment and for the products you've mentioned, I'll suggest your to start looking at browsers plugin development and to start understanding the HTTP protocol and all the related technologies around it ( HTTPS, encryption, authentication etc. )
The public API let you easily login into an account, but you don't really need one to do it ( although it make life much more simpler ) If you do not have a public API, you can still login into any account by "simply" doing what the browser does. Sending an HTTP(s) request with the appropriate security mechanism and following the protocol.
If you know how does the browser send the request and you have the user trusting your their passwords, the only remaining thing you have to do is ... :) code it.
As of now the question is too broad to be answered. Pick one service at a time and ask specific questions about it.
I would suggest you to start with the previously mentioned "Google Accounts" API and learn from there.
One open source product that already manages google account authentication is "Ubiquity" you can take a peak at their source code and start understanding how do they fetch the user contact list.