As of now (22 Sept 2009), there are 3 "memtests" that I know of
This is a program that runs from within windows and tests memory. You can get the free version for free obviously. It's a nice tool to have, but I don't know how accurate it is, even though there's been somebody on a forum who said that this program found errors in his memory when memtest86+ failed to (this actually seems to happen to one of my PCs as well, but I haven't tested it extensively). Also when people refer to "memtest" they sometimes do not refer to this utility, but rather "memtest86" or "memtest86+".
This is the original memtest86, and the program that most people refer to, for testing memory. You can boot off of it (either from a CDrom or floppy), and it runs a bunch of tests to evaluate your memory. There has been a period when this program was not updated, for 2 years or so, and memtest86+ was developed (by another team) to compensate for that. This does not hold true right now, though, since the last version as of today is v3.5 (released at 3 Jan 2009).
This is the program that was developed when memtest86 was not being actively updated. However, both are now being updated, and have their own version numbers which can be confusing. As far as I know, these programs run the same tests (as of today, 22 Sept 2009), and therefore it doesn't really matter which one you choose (aside from possible bugs and/or compatibility issues). Also keep in mind that the memory tests themselves shouldn't change too much with every version, what changes is hardware compatibility and bugs. Latest version of memtest86+ is 2.11, released at 22/12/2008.
Here's a quote from wikipedia:
Also, if you're using either memtest86 or memtest86+ it would be a good idea to leave them running for at least 15 hours, because in my exprerience I have seen the first error after the 30th pass of the standard set of tests (1-8), which translates to around 12 hours (depending on hardware configuration). I know, though, that this is an extreme exception (and probably occured due to excessive heat, as I was running on a laptop). Generally (as mentioned in the readme), the first pass should detect all but the most obscure errors.Memtest86 was developed by Chris Brady. After Memtest86 remained at v3.0 for two years, Memtest86+ was created by Samuel Demeulemeester to add support for newer CPUs and chipsets. Both are now (as of 2009) being actively worked on. The bootloading code was originally derived from Linux 1.2.1. Memtest86 is written in C and x86 assembly. The source code is released under the GNU General Public License (GPL). The current version of Memtest86 is v3.5, released on January 3, 2009 and v2.11 of Memtest86+, released on December 22, 2008. Both versions now support current dual- and quad-core-CPU's and the corresponding chipsets. The latest release of Memtest86+ supports Intel-based Macintosh computers.