It really depends on what technology restrictions you have.
One way is to have a spatial database with the outline of the countries and cities you are interested in. By outline I mean that countries and cities are store as the spatial type polygon. Your set of coordinates can be converted to the spatial type point and queried against the polygons to get the country/city name where the point is located.
Here are some of the databases which support spatial type: SQL server 2008, MySQL, postGIS - an extension of postgreSQL and Oracle.
If you would like to use a service in stead of having your own database for this you can use Yahoo's GeoPlanet. For the service approach you might want to check out this answer on gis.stackexchange.com, which covers the availability of services for solving your problem.