Visa used to have three systems called Visa (for credit cards), Visa Debit (for debit cards), and Visa Electron (also for debit cards but only launched in select countries). When they launched their marketing campaign 'Go With Visa', they eventually simplified this so that most cards on their network these days - whether they are credit cards or debit cards - carry simply the Visa logo, and pass through the same payment processing network. Thus, as of now, a Visa card is one of the best bets you have. Still, there are caveats - a Canadian friend who came to Singapore found that he could only use his 'Visa Debit' logo card on select ATM networks that specifically supported it.
MasterCard has the main brand, and another one used only with debit cards called Maestro (again, only launched in select territories). Unlike Visa, an establishment or an ATM that can process MasterCard logo cards is not necessarily able to process Maestro (unless it supports Maestro too, of course).
In addition, many countries have their own payment networks that are massive in their own country but accepted nowhere else. China, for instance, has UnionPay which I think is the only domestic card allowed in China; it's also supported by quite a few Vietnamese, Malaysian, and Singaporean banks. Singapore too has its own system called NETS which until a few years ago was the only card you can use anywhere in the country, but that has changed now. Most of Asia has fairly good coverage for Visa and MasterCard now, with the former far more popular. At many merchant locations in Asia, you can only swipe a Visa card. Most bank ATM networks do support MasterCard, however.