Calculating polygon areas in QGIS?

Make the layer editable, then use the field calculator (Layer>Open attribute table>Field Calculator/Ctrl+I or right click shapefile>Open attribute table>Field Calculator/Ctrl+I). There is an operator "$area" that will calculate the area of each row in the table. All units will be calculated in the units of the projection, so you probably want to project it to a projection that uses feet or metres before doing that, rather than lat/lon.

This can also be done with Vector|Geometry Tools|Add/export geometry columns, which creates a new shapefile with area and perimeter (or length) columns added.

Edit: (using the tool above, you can also unselect "save as new shape-file" in V1.8, the shapefile is now only updated!)

Using the field calculator is probably a better idea, though, as it doesn't require the creation of a new shapefile.

I wrote a script specifically for this. If you don't want to reproject your data, you can compute the area using ellipsoidal math.

Processing Toolbox -> Tools -> Get scripts from on-line scripts collection -> Ellipsoidal Area

You will find the script installed in Processing Toolbox -> Utils -> Ellipsoidal area

The tool should be self explanatory and will allow you to calculate area in units of your choice regardless of projection.