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What is the difference between the Geographic coordinate, transformed coordinate(projected) ,cartesian coordinate, and screen coordinate? Is there a live example?

And what is the association between them? For example, on geographic coordinate will have one or more transformed coordinate?

The following is a .prj file from this post:

PROJCS["Lambert Azimuthal Equal-Area",
       GEOGCS["GCS_WGS_1984",
              DATUM["D_WGS_1984", SPHEROID["WGS_1984", 6378137, 298.257223563]],
              PRIMEM["Greenwich", 0],
              UNIT["Degree", 0.017453292519943295]],
       PROJECTION["Lambert_Azimuthal_Equal_Area"],
       PARAMETER["false_easting", 0],
       PARAMETER["false_northing", 0],
       PARAMETER["latitude_of_origin", 0],
       PARAMETER["central_meridian", 20],
       PARAMETER["xy_plane_rotation", 0],
       UNIT["Meter", 1]]

Then what confused me is that Lambert Azimuthal Equal-Area is the name of the projection coordinate system, and it specified the geographic coordinate system here(GCS_WGS_1984),why? Does it mean that this projection is only fit for the GCS_WGS_1984 geographic coordinate? As far as I understand, the projection have nothing to do with the geographics coordinate.

For example,given a geographics coordinate system GCS_WGS_1984.

Now I can use different type of projection(like UTM, and Lambert Azimuthal Equal-Area in the post) to transfrom the geographics coordinate to projected coordinate. So the projection is not bounded with the geographics coordinate system. Isn't it?

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    Your title contains some good search terms: they turn up loads of relevant answers. – whuber May 14 '13 at 00:30
  • I am sorry but I have read the post and other docs I can got from google. But I can not make it clear enough. For example, does the projection have nothing with the geographic coordinate? I mean does the geographic cooridnate system will have a related projection? – giser May 14 '13 at 00:42
  • Please take some time to read through the search results. After going about a third of the way through I found http://gis.stackexchange.com/questions/2769, http://gis.stackexchange.com/questions/7072, http://gis.stackexchange.com/questions/21554, and http://gis.stackexchange.com/questions/8430 appear to have helpful information. I'm sure you can find more. Literally thousands of questions here deal with the details of working with projected and unprojected coordinates, so there is no lack of examples. – whuber May 14 '13 at 00:51
  • @whuber: I update my post after I read the posts. Can you spare some time to have a check? – giser May 14 '13 at 01:35
  • The duplicate explains the difference between a projection and a datum. The (conceptual) datum in the .prj file is everything specified within the GCS record. (The DATUM subrecord is actually just a part of what we understand to be a geographic datum. The other subrecords, PRIMEM and UNIT, stipulate an origin and a unit of measurement, respectively.) The other records in the .prj file describe the projection from that datum into a projected coordinate system. Collectively they indicate exactly how any point on earth receives (x,y) coordinates. – whuber May 14 '13 at 01:39
  • I read the PROJCS each words one by one. And it seems that I take the PROJCS the same as the projection. It seems that the projcs(projection coordindate system)=gcs+projection(something like the Formulae to make the transformation)? – giser May 14 '13 at 01:57

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