I'm using QGIS 2.6 and regularly update a shapefile layer to indicate felling progress. However, if I edit the nodes of a polygon, save the layer out then look at the Attributes table the area shown doesn't change.
The identify tool provides a quick way to access area measurements. You can change the units by going to:
Project menu > Project Properties | General > Measurements
WARNING: Sometimes area calculated in QGIS is wildly inaccurate. This can happen regardless of whether you use the identify tool or the field calculator in the attribute table. It's usually a result of having multiple layers with different coordinate reference systems in the same project. If you start seeing area numbers that don't make sense, try loading the layer into a blank project, turn off on-the-fly reprojection, and redo the calculation using field calculator.