For reference, the machine I am on is a Mac Pro, 12 GB RAM and running OSX 10.7.3. It has 2 X 3 GHz Quad Core Intel Xeon processors so it should make short work of anything I through at it. My machine is Unix based so please modify the instructions based on your operating system. Linux users will be roughly the same.
1. Grab your browser (Chrome of course) and point it at http://neo4j.org/download/
2. There are a variety of options open. Unless you want to start with the über - enterprise version or build Neo4J Community from GitHub source, simply grab the latest stable version. In this case it is the Community (ie "free") version 1.7. If you have a slow connection, use the time wisely to view Emil's video while the download completes. You will have to choose a location for the download.
3. Unpack the source with whatever tools your Operating System provides. Once unpacked, it will look something like this: