What is the easiest way to check for plagiarism in student essay papers?
There are plenty of tools out there. The best 'seems' to be TurnItIn. It's costly so if the school won't pay for it, you might not be interested. My school does not so I found other options.

Here is a tool called WCopyFind that is very good at finding plagiarism when one student copies from another but it will not find something online. After collecting papers for several semesters it has been catching more and more and most students take the same easy way out.

There are other options if you search for online plagiarism checking but they are not as robust as this tool above...however to really see the benefits, you need to maintain a library of old papers to compare against. What I like about it (besides being FOSS) is that it is highly configurable (consider a match 6 words in a row, or consider a match 9 words in a row where 70% of the words match). It all runs local (since there's no internet search) so it's quite fast.