On a number of the site templates such as Team Site, you will find that you get an Announcement web part to begin with.  After you have used the Announcement web part that the SharePoint template provides for you, you may decide you have another list of Announcements to add. 

So as the site administrator, you go to Site Actions >> Edit Page.  You then figure out which web part zone you want to add your new Announcements web part to and click “Add a Web Part”.   In the “List and Libraries” section, you simply click Announcements and hit OK to add this as a new web part.

Seems simple enough, right? …………..  WRONG!  The web part you have added is tied to the same announcements list as the first one.

This will actually duplicate the default Announcement web part that the site template provided for you.  You need to take a couple of different steps to add a different Announcements web part to the same page.  What you need to do is to create a new announcements list first.

Go to Site Actions >> Create

Choose Announcements from the Communications section.

Give it whatever name suits the purpose of the announcement.  For this demonstration, I will name it Announcements2.  Click Create.

Now go back and go thru the original steps I described earlier. 

Site Actions >> Edit Page >> Add a New Web Part. Now I can see both my existing Announcements and my new Announcements2.  Choose your newly created Announcements2 or whatever you named it and add you content.

 
For a future reference, when you’re adding a web part, the List and Libraries section is directly linked to any lists or libraries you have created within your site.  They will be there if you choose to show them as web parts.  The world of SharePoint makes sense again!