Is there a way for me to share a map that I created, using ArcMap, with others through something like DropBox?
The simplest, if you want to ship everything unpackaged, is zip the folder containing your mxd and data in a transportable project and use OneDrive, DropBox, CloudyWhatever to share the whole unpackaged package.
I suppose, you really need to specify what you want to share. A screen grab of a map embedded in an email is not to be excluded either.
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