Actually, NHL isn't the only hockey in the world. Yeah, the British league is more or less a beer league (sorry, no offense meant!), but there is the KHL (mostly Russian teams with some international ones, there will even a Chinese team be soon), Sweden, Finnish, Czech and some other leagues, which are the best of the rest. The KHL is usually considered the 2nd best in the world (behind the NHL, of course).
There is an international hockey as well, like Olympics, World Hockey Championship and a lot of junior tournaments, which all are held under IIHF (international federation, like FIFA, but in hockey) rules. The most interesting thing about international hockey is it's mostly played on international size rinks, which are the same length as in the NHL, but much broader. That's why international hockey looks different, than NHL one. And even NHL players play different on the bigger ice. It's slower, less tough, fewer shots with more puck control and passes.
When European players come to the NHL, it usually takes some time to get used to new hockey for them. The best players do it pretty well, though. For example, the Blues have a few European players on the roster, and the team's main star is Tarasenko, who's Russian and came to the NHL from the KHL a few seasons ago.
Chicago, Blues are playing with right now in the playoffs, has Panarin, who's another Russian and the KHL player (and Tarasenko's best friend, btw). He's been playing his first NHL season and already become top10 scorer, so yeah, the difference between international (or European) and NHL hockey isn't like the difference between American and international football. They are pretty much the same game, but with nuances.