A cross-listing can be a direct secondary listing of the ordinary shares or a depositary program such as an adr or gdr. It brings a second regulatory regime, a second set of disclosure obligations, and a second sedol, but usually one isin.
For traders the main effect is time coverage. News breaking while the home market is shut gets priced on the foreign line first, so the overseas close is often the best available estimate of where the home market opens.
Example: a company's home market closes at 11:30 New York time. A regulatory decision lands at 14:00 and the US line falls 7%. The home market gaps to roughly that level the next morning, having never traded on the news.
Related: dual-listing, adr, gdr, isin, sedol