You have a couple different options. What you are experiencing when importing in to ArcMap is called "project on the fly". Since your ArcMap has a raster dataset with a projected coordinate system, your dataframe assumes that coordinate system. This allows your .shp file to align properly in your map when importing.

In order to project your file you can use either the Define Projection Tool in ArcToolbox -->Data Management-->Projections and Transformations...or you can right click the shapefile in catalog, choose properties and either Select a predefined Coordinate system or import from another existing dataset.

Since you are trying to calculate the area of polygons, you will likely want to choose a projected coordinate system that is already in the units you desire your area calculation to be formatted in. I recommend using a State Plane coordinate system related to the area you are analyzing. Choose Coordinate Sytems-->Projected Coordinate Systems-->State Plane--> and choose your desired datum and unit.