The closest international airport is Düsseldorf airport (DUS), and this is probably your best choice to come here from abroad. From there, you can reach Essen conveniently by regional trains (in about 30 minutes, about 10 EUR per trip). There also are fast trains (IC/ICE) which are a little faster, and more expensive.
From Cologne/Bonn airport (CGN) you can reach Essen in a little more than one hour (with long-distance IC or ICE trains, about 25 EUR per trip) or one and a half hour (with regional trains, 15 to 20 EUR). Possibly you will have to change trains in Köln (Cologne).
Finally, you could consider flying in to Frankfurt airport (FRA), where many international flights arrive, but which is located quite a bit farther from Essen. From Frankfurt airport, you can reach Essen by train in about 2 hours by ICE trains, for about 80 EUR. There are some direct connections; otherwise you will have to change in Köln (Cologne).
Please note that tickets for regional trains are not valid in ICE, IC and EC trains. For the latter, you need special (and more expensive tickets). The ticket vending machines show the available connections, and it's easiest if you buy a ticket for the exact connection you are going to use.