A datum is essentially what you call an approximation of the Earth's shape and surface so that it comes out as an ellipsoid (see e.g. nice illustration above from here: the 'geoid' is what you get if you simplify the global surface by ironing out all valleys and mountains, and the ellipsoid is a further simplification of that taking out the lopsided-ness of the globe too).

Once you have the datum, you can specify longitudes and latitudes on the surface of that ellipsoid. These are called GCS coordinates (geographical coordinate system), e.g. coordinates for a control point in Redlands, California, are 117° 12' 57.75961'' W, 34° 01' 43.77884'' N (according to NAD83). n.b. because no datum shape is a perfect sphere, the longitudes and latitudes you get for a particular location depend slightly on the datum used, e.g. that control point in Redlands has coordinates 117° 12' 54.61539'' W, 34° 01' 43.72995'' N according to NAD27 (Kennedy & Kopp 1994:p.6)* .

* I have been told that the datum shift between using different datums (i.e. the horizontal difference between the coordinates of a particular location X according to two different datums) is never more than a few 100m. So, I thought to search for a map of where the shift is greatest, but I couldn't find one (just a lot of learned blogs like this one rather snootily explaining that there is little point in looking for something like this (??)).

Once you have specified a location on the global surface uniquely (i.e. you have the GCS coordinates), then you are not quite finished. For the purposes of assembling a GIS map, you need to decide how to 'project' it, which means using a projection transformation to create corresponding PCS coordinates (projected coordinate system) for each point (Kennedy & Kopp 1994). You can usually expect that the PCS coordinates will be in metres or something similar (and are called "easting"s and "northing"s rather than "longitude"s and "latitudes", e.g. UTM coordinates (a PCS) for New York are 589912 m E , 4509398 m N).

Finally, be aware of the term CRS (coordinate reference system), which refers to any kind of coordinate system (GCS or PCS). Generally speaking, data files can be found where the CRS is a GCS (most usually with dimensions in units of degrees) and others where the CRS is a PCS (most usually with dimensions in m or km) and many files where the CRS has not been specified at all.

OK: All happy with that?

Well, personally I think that you shouldn't be, because there are several gaping holes in this explanation above (!). Just for starters, what does it mean a "projected" coordinate system when you can take the two numbers from any GCS and plot those on a piece of paper anyway (i.e. in a mathematical sense, all GCSs are "projected" too - in the sense that they are a mapping from 3-space to 2-space - as you can see from the Kelowna map right where grids for both UTM coordinates (a PCS) and long/lats (a GCS) are plotted on a 2D surface so clearly both are "projected" in that sense)?

Let's rewind just a little and ask why we bother with the UTM coordinates on a map like the Kelowna map right? Why not let the computer handle those in the background and we will just use the long/lats?

The answer is surveying. Above all other things, surveyors need to know distances and areas on a map. If you need to know the distance from Kelowna to Rutland on this map, then you can get an approximate solution using the long/lat coordinates and the Haversine formula, but that assumes the globe is a perfect sphere (i.e. not datum differences) and is not sufficient for surveying. Using the UTM coordinates, however, calculating distances and areas is just a case of applying Pythagoras (see e.g. here). So: we use PCSs like UTM in order to be able to calculate distances and areas precisely, NOT because you can't display a GCS map on a 2D surface.

Understanding this, you can see (I hope) that several error messages you get from GIS packages are actually really quite misleading, e.g.
   - If a GIS package says that a layer you have imported "cannot be projected" (e.g. "data can be be drawn in ArcMap but cannot be projected" from ArcGIS), this doesn't make immediate sense (because any list of coordinate pairs can always be drawn, and therefore projected!): what it means is that the information on how to project it are missing from the input file (no, I don't know why they don't say that!).
   - If your layer has been loaded into a GIS package and it displays at the wrong location, the GIS package will generally inform you to check the CRS of that file and tell you to consider "reprojecting" it. Personally, I think an error report saying "CRS of input file does not match CRS of map project" would be more helpful because in my experience the input file is usually OK and this message would point the User towards checking the CRS of the map as well (which is often set by default and can easily not be what the User thinks it is).

A final point: when I started with GIS I thought that I would have freedom to choose the final projection of my map to be whatever I wanted (e.g. assemble all the details of my map, and then at the last minute flip it on to Gall-Peter's projection simply because I prefer it, rather like you can prepare a document in Word and then flip the margins, page sizes and fonts at the end). GIS packages are (as of 2024) universally not set up to do this:
     (a) You need to choose your final projection at the start (the 'project CRS') and then stick to it,
     (b) You need to carefully 'reproject' every layer that doesn't conform to the project CRS when you load them in (and don't even think of trying to convert to such an unusual CRS as Gall-Peters unless you really need to, because other people will almost certainly not be able to use your map later) and
     (c) If you want your project CRS to be a GCS rather than a PCS, then check that you can measure distances on your map: in ArcGIS I always got things like the distance between X and Y being given in degrees (i.e. using the 'map units' - the project CRS - which only makes sense inside a GIS package), but I tried this out in QGIS and it gave me the option to have distances in m or km. Just because of this, I'm now moving over to QGIS and abandoning ArcGIS (!).
​In summary, you don't have that freedom (yet!).

More information: see here, here and/or here.
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Map of Kelowna, BC, Canada (from here).

Kennedy M & Kopp S (1994). Understanding Map Projections. ESRI.