To determine which projection, if any, is currently defined for a layer, double-click that layer's name in the Table of Contents to bring up the Properties dialog. Look at the Source tab.
You can georeference, or register, raster imagery such as satellite or aerial photos, or scanned paper maps, so that ArcGIS will know its location and allow you to overlay other GIS layers. This involves matching control points in your raster image and on your already-georeferenced target data. This blog post outlines the process in easy-to-follow steps. Further online help from Esri.
Projection: The mathematical model for flattening the rounded Earth. There are many, and the best one to use depends on scale, location, etc. "Unprojected" means that the layer map uses latitude and longitude coordinates (degrees/minutes/seconds or decimal degrees). Online help.
Coordinates: The grid used determine location on a map. On projected maps, distance will be measured in meters, feet, etc. from defined meridians and parallels. Lines of longitude aren't parallel, so projected coordinates must be used for precise distance and proximity calculations. Online Help: Geographic (unprojected) coordinates | Projected coordinates
Datum: Relates to the model (the spheroid) being used to approximate the Earth's shape (it's not spherical), and there will be a defined point where the surface of the model intersects with a point on the real Earth (the datum). Online help. | More on datums.
The Data Frame's coordinates will be set by the first layer that you open.
For visualization (doesn't affect the projection of the underlying layers), once everything is defined you can change the look of the map on the fly in the Data Frame properties. The projection/coordinate system doesn't need to be the same as the layers, as long as everything is defined.
Useful discussion of issues surrounding Coordinate Systems and Data Frames. Helps tie the concepts together.
Geographic transformations: Choosing a datum transformation method is sometimes necessary when using the Project Tool. A zipped Word file lists many of the other transformations that might be used for other areas.
Probably the most common transformation will be WGS84 to NAD83 or vice versa. For going either direction, choose the datum transformation "NAD_1983_To_WGS_1984_1" which works well for the North America, defined as "Canada, Central America, Mexico, and United States (Alaska, CONUS)" (CONUS = Continental United States).
Census Bureau 1990 and 2000 Shapefiles: These all use unprojected latitude/longitude coordinates (geographic coordinate system or GCS), decimal degrees, 1983 North American Datum (NAD83), but the information isn't stored with the shapefiles in a .prj file. Use the ArcGIS Define Projection tool to fix this. Census boundary shapefiles for 2007 to present do include coordinate information with the files.