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I'm exporting data from a program into a .csv for RStudio, which will then go into ArcMap.
I haven't done this before and I've been given two projection options which are:
What is the difference between the two of them and how do I know which one I need to use?
EPSG 4326 is a coordinate system of latitude and longitude based on an ellipsoidal (squashed sphere) model of the earth. Its not really a projection.
EPSG 32637 (and any of the "UTM Zone" projections) is a projection of the round earth onto a flat surface, and has coordinates in metres. Its only accurate within the area of the zone, which is a range of longitudes depending on the zone number. Calculation in EPSG 32637 such as distance and direction can be done using flat geometry like Pythagoras' theorem for distance and simple products for area. To truly compute distances or areas on the curved surface of the earth you would use EPSG 4326 and more complex formulae for distance and area. ArcMap can probably deal with either, as can the spatial packages for R.
Its impossible for us to say which one you need to use since it depends very much on how much precision you need and what you are going to do with the data anyway.
I read your question as deciding which of the two given spatial reference systems apply to your CSV coordinates, fortunately this is a fairly easy decision.
As geographic or projected are your only two options look at the numbers given for the coordinates, if they're less than 365 x 90 (a circle in the X then half a circle in the Y from -90 to 90, which is frequently written lat, lon which is Y,X) then it's WGS84 Geographic (EPSG:4326), if the numbers are much bigger than that then your numbers are in UTM coordinates, thus WGS84 UTM Zone 37 North (EPSG:32637) is your only option.
An example of geographic coordinates: 173.847431,-35.516882 (X and Y, which makes it 173°50'50.753"E 35°31'0.774"S) which would be often encountered as -35.516882,173.847431 as lat/lon. This example point is somewhere in the north island of New Zealand - I don't have any northern hemisphere data to grab an example from.
An example of UTM coordinates: 758223,6065904 (East and North).
As was mentioned before by Spacedman geographic coordinate pairs are unique for the whole globe and normally have an origin in the ocean west of Africa (Null Island it's called but there's really no land there) so you can find a unique value for any place. Conversely UTM coordinates are not unique over the whole of the Earth, just within a particular slice of the earth. Each UTM zone is 6 degrees wide and also are broken into north and south which makes 120 UTM coordinate reference systems, as well as innumerable Lamberts, Albers, Polar etc. projections that a single coordinate pair may exist upon, all based on WGS84 datum - and there are many more based on different datums.
So as you can see it's very convenient being limited to just the two options.
