I am searching for a web-based service that will display a map
customized to your specifications.  I want to select a country, enter
or select the cities I want to locate (must be able to select multiple
cities), and end result will be a map of that country with only the
selected cities showing.  Needs to be an image I can copy/paste into
another application.  Does anyone know where I could find such a
service?  I have software that does similar things but it is very
difficult to use - need something quick and easy and preferrably web
based.

Google Earth has now been released as a free public beta:

http://earth.google.com/

Am not sure of the precise functionality of this version but it may
fulfil your requirements.

While Google Earth is definitely impressive, it doesn't have the
functionality I need, i.e. to be able to "pin" multiple selected
cities at once on the same map.

The 'Plus' version of Google Earth will allow you to pin placemarks on the map.

http://www.nationalatlas.gov  has a nifty map-making function that
might serve your needs as well.