ArcMap Desktop Introduction: Projections and Georeferencing (Lining Up Data)

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.

View ... Data Frame Properties ... Coordinate System tab.

The Data Frame's coordinates will be set by the first layer that you open.

• The various possibilities are displayed in tabular form in a Quick Reference Table.
• If all layers have defined projections/coordinates, then they should all overlay properly.
• If the first layer is defined, but a subsequent layer is undefined and using different projections/coordinates, to get them to overlay you should just need to use the Define Projection tool (see below) on the layer(s) with unknown coordinates.
• If the first layer is undefined (unknown coordinates), then to get layers in other projections to overlay, in addition to using Define Projection on it and any other undefined layer, you'll need to change the Data Frame's coordinate system in the dialog.
• If the first layer has undefined geographic (latitude/longitude) coordinates, then ArcGIS discerns degrees and the cursor's position will display as degrees, but the Data Frame properties will still show an unknown coordinate system.  Subsequent unprojected lat/long layers will overlay properly, but projected layers won't. To get them to overlay, you must define the unknown layers' projections, then also define the Data Frame's coordinates.

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.

• Define Projection: This tool allows you to tell ArcGIS what projection/coordiates/datum a layer is using, if such information wasn't included with the layer (layer Properties shows Unknown projection under Source).  You can also define the projection in ArcCatalog.  Online help.
• Project: This tool allows you to change the projection of a layer, or to project an unprojected layer.  It creates a new, projected, layer (doesn't alter the original). This procedure is necessary for using many of the geoprocessing tools, such as those relating to distance or nearness analysis. Online help.
  
  When you choose the Output Coordinate System in the Define Projection or the Project tool dialog, you can pick a projection several ways, including
  • Select:  Drill down through lists of Geographic or Projected coordinate systems to choose one.
  • Import:  Pick another layer that's already projected the way you want to choose its projection.

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).