Define projection and project in ArcGIS

To understand the projections, it is necessary to know: What is the Projection and Datum?

Projections are transforming the area or the reference ellipsoid on a flat surface, representing the globe on a map, with the help of a network of meridians and parallels through a grid, there are cylindrical projections, conical and azimuthal each with their distortions, to avoid ideally represent the globe on a sphere, but not practical. So it’s a complex issue agree on selecting the best projection, so each country in the world uses its own coordinate system based a specific projection.

The Datum, first of all it must be clear that it is the geoid and ellipsoid, the first refers to the theoretical surface of the earth connecting all points of equal gravity, while the ellipsoid, as we all know that the earth is not round and as we were taught in school that their poles are flattened, so the ellipsoid is a mathematical model based on the crushing of the earth and its major and minor radius, rotating on its own axis, adapting to the shape of the land in a particular area.

So stressed above, the datum is the tangent to the ellipsoid and geoid point.

Now we pass the theoretical to the practical, considering two cases to solve in ArcGIS.

• First case: When I have a shapefile (shp) or raster, without defining its spatial reference (define projection as long as the origin of its known spatial reference). For this we use the tools ArcToolbox, and select the tool below:

ArcToolbox > Data Managements Tools > Projections and Transformations > Define Projection

Within this tool simply select the shapefile or raster and to establish the corresponding projection.

• Second case: The shapefile or raster has a spatial reference, but is not required (must change the projection). In this case we need to project the new coordinate system. For shapefiles -> Project, and rasters -> Project Raster from the ArcToolbox:

ArcToolbox > Data Managements Tools > Projections and Transformations > (Feature) > Project
ArcToolbox > Data Managements Tools > Projections and Transformations > Raster > Project Raster 

Fill in the following fields:

• Input Dataset or Features Class: Shapefile (raster) to be projected.
• Output Dataset or Features Class: Directory path to save the shapefile (raster).
• Output Coordinate System: Select the new coordinate system.

• North America: NAD27, NAD83 y WGS84
• South America: SAD 56 y WGS84
• Argentina: Campo Inchauspe
• Brazil: SAD 69/IBGE
• Spain: ED50, ETRS89 since 2007 throughout Europe.

Please help me with writing (comment), my English is not good.