The first thing you should probably understand is that portlets are not servlets. Even though, portlet containers may use servlets underneath portlets (which I'm not even explicitly sure about), the programming model is different and you will need to mindful of that when designing and implementing solutions with them.
Portlet applications can be directly referenced by the context root you defined when you installed it. However it's the portlet container that what will call your doView, processAction, etc methods and help maintain the state of the portlet and the navigation state of the user. Adding the portlet to a page let's Portal know where you want the portlet to be available in the page model.
You have two administration consoles because WebSphere Portal runs on top of WebSphere Application Server. That is the standard WAS admin console you're using there. You'll use it to administer your standard WAS stuff (servers, database connections, web services, etc). The Portal Admin console is there to administer portlet specific specific things, such as the page model, portlets, themes, etc.