I want to crop and print a raw file that is 5184 x 3456. I am using PS Elements to edit. I used the crop tool (aspect ratio) and set the dimensions to 20 (w) x 16 (h). Nor sure what to set the resolution to. I don't understand how to correctly resize/crop a photo for printing by a local retail printer (Walgreens). Should I be using the resizing tool also. What is the difference in the 72, 300, and 600 dpi?
It really could not matter less what the dpi is, in that if you send some pixels to the finisher and ask that they print it 20x16 inches, they will print it 20x16. They must ignore any dpi value that might be in the file. Because 3456 pixels / 16 inches is 216 dpi, which is the only way 3456 pixels can come out 16 inches. They will take care of doing that. The value of dpi is unimportant, other than it should be sufficient.
However, the way Adobe used to be, if you use the Crop tool (to make it the necessary 20x16 shape), and if you also enter a dpi in that Crop tool, then it will resample to make it be that size at that dpi resolution you asked for. But if no dpi is entered (dpi left blank), it would just crop the shape, which is what you want.
For maximum resolution at 20x16 inches, you want your cropped 5184x3456 pixels to come out about 4320x3456.... cropped to shape, but still 3456, no resampling.
If you print at home by selecting the Print menu, without selecting any other resize options, then that is the only time the dpi value in the file is used. Then it will space the pixels at the X pixels per inch on paper. But if you tell the finisher AxB inches, that is what they will necessarily do, at whatever dpi the pixels will support.
There are several parts of this question. There are some similar questions out threre, so I'll be more specific.
1) Your Data:
5184 x 3456 px.
20 x 16 inches.
Note that I added the actual units.
This is important. Resolution is a relationship between 2 units. 1 unit is of information (A), one is for dimension (B).
In this case the unit "A" is pixel and the unit "B" are inches. Unit B can be for example centimeters.
2) The proportion. You define that using the crop tool.
20 x 16 = 1.25. The width is 1.25 times greater than the height.
So 3456 x 1.25 = 4320. You need to crop your image that size.
From there you do not to do anything. The file size is good enough. But here are some more numbers.
4320 px on 20 inches (the units again) are 4320/20 = 216. This is your photo's resolution printed at that phisical size. 216 pixels on each inch... pixels per inch or ppi. UNITS AGAIN!
3) 72, 300 and 600 are some ppi "standards".
The 300 ppi is the most important "standard for comercial printing, like a magazine. But even this one is not carved in stone.
600 ppi is for a special case of image that is only pure black and pure white. No gray or color, that is called a monochromathic image.
72 ppi... That is a pointless historical information that only confuses web designers.
There are other values. But that is an offtopic for photography. That is a designers issue.
Returning to the Photographers point of view. A photo will be fine if it is on the 150 - 300 ppi.
You do not need more than 300. But sometimes, where you are printing wall sized photos, you could get away with lets say 50 ppi or less.
The correct unit for you is ppi, not dpi.
The dpi is another diferent relationship to mesure a printer's resolution, not a photo resolution. A printer is defined on how many dots of ink can deliver on an inch... dots per inch.