MemTest86+ is a simple but thorough tool (based on the original MemTest86) which will scan your PC's RAM for errors. It arrives in the form of an ISO image, so isn't quite as convenient to use as some of the competition, but once you've burned it to disc or a USB key - which won't take long, as it's a tiny 1.75MB in size - everything else works very well.
MemTest86+ boots immediately, for instance. There's no complicated operating system which need to fire up, first - you're straight into the program's simple text-based interface. A menu offers a selection of test options, but you don't even have to worry about that: just wait a few seconds and the default test pattern will begin.
The screen then updates to show you exactly what's going on, the various tests that are being used. It's fairly technical, but if nothing else the "Pass %" and "Test %" will show you how far through the checks you are. And again, there are a lot of error types which could be reported at the end of the test (MemTest86+ is very thorough and detailed), but you don't really have to understand those, either. If you see any errors, there's a problem with your hardware; if you don't, your issues are probably somewhere else.
Should you use MemTest86+, rather than the original MemTest86? It's certainly worth a try; this new edition supports many more CPUs, improves SPD and memory controller detection, and adds a few useful extras (the report includes your motherboard manufacturer and model, CPU temperature and more).