Roman or Latin script is the one used by all the Romance, Germanic, Western Slavic, Ugrofinn and Celtic languages + some of the Southern Slavic, Asian and African ones.
Some languages, despite not exchanging their own script for the Latin one, have accepted a romanization system mostly as a tool for learners and technology. (Japanese rÅmaji, Chinese pinyin and others)
It is based on the classical Latin alphabet and you can read more about its develpment on wikipedia.
While most languages using the script use the whole set of basic characters, many of them have adapted it to their own needs using diacritics.