How is it possible to display an announcement list of items, in two columns in the same web part, with just their names as clickable links being displayed on the web part.
Definitely. Have a look around the web for DataViewWebParts (or the new XsltViewWebParts if you're using SharePoint 2010).
You'll be using SharePoint Designer for this - basically you point a DVWP at your announcements list, and you customise XSLT (eXtensible Stylesheet Language Transformation) code to present the list in any way you want.
Out of the box, the easiest way is to create a boxed view:
- Create a new standard view
- in the fields to display, only select Title (linked to item)
- under Styles select one of the boxed views