I have added XY coordinates on map layers in ArcMap.
These coordinates now show up as points.
However, I want to be able to color each coordinate point a different color.
Short answer: Go to this layer's symbology and under Categories pick a unique Value Field, push "Add All Values" and OK.
Long Answer: In the Table Of Contents (the window on the left that contains the XY points layer), right click the layer of XY points and push "Open Attribute Table". Here you need to find a field with a unique value for each row, which mustn't be the first field because you can't color based on it. If there isn't one, you need to create one. Push "Table Options" (upper left) -> Add Field -> under Type choose Long Integer, push OK. Now right click the new field -> Field Calculator. Under Fields, push the first field - should be OBJECTID or FID, or OID and push ok. Now right click the layer in the Table Of Contents -> properties -> Symbology tab -> Categories -> Unique Values, Under the Value Field, find the new field, Choose the right color ramp under Color Ramp and push Add All Values.