We'll start with an out of the box Team Site.

The trick is in the Ribbon. We have to have it be contextual to editing the page. To do this we simply click on the Edit Page Icon.

Click on the insert tab to make the ribbon icons contextual to inserting a web part.

You should see the icons: Web Part, Existing List, and New List. We will choose Existing List.

We will click on Tasks in the Web Parts section to select the Tasks List. In the About the Web Part section we see that the Add Web Part to: Rich Content is Selected which means this is the area of the page the Web Part will be located in.

After a moment we will see the Tasks Web Part Inserted into our page. Now all we have to do is click the save icon at the top of the Ribbon.

That's it…. A couple of things though, if you don't see the list you are trying to insert it's very likely that it doesn't exist. This happens all the time with List web parts. The list has to exist on the Site for you to be able to point a web part at it. There are other things as well but this is the most common. That's how you add a web part the front page of the team site in SharePoint 2010.