MakeHuman downloads (upcoming 1.2.0 release)
At this point, windows users will want to download the nightly build (see which version should I download?). The nightly build can be found in the download directory. Look for the file named *-nightly-windows.zip.
If, for some reason, you do not want the nightly, you could try downloading the latest beta of the upcoming 1.2.0 release. Please read at least the "known issues" section on the linked page before downloading.
Linux users are primarily encouraged to run from source, but there is also a PPA for 1.2.x at https://launchpad.net/~makehuman-official/+archive/ubuntu/makehuman-comm...
MakeHuman downloads (old version)
If you want to download the source code for MakeHuman and/or run from a source snapshot, see the MakeHuman repository on Github.
You will most likely also want some of the commonly used plugins, such as MHX2 and the asset downloader. These can be found on the plugins page. These are bundled in the nightly build.
Extra assets such as clothes, hair and skins
More assets for MakeHuman can be found in the online repository.
You can found old stable releases at http://files.jwp.se/archive/
Related & useful
These tools are not part of MakeHuman, nor managed by MakeHuman, but are often used in the same pipeline.
Blender is an open source 3d modeler, ray tracer and a whole lot else. This is the 3d software/environment that will work best together with MakeHuman
MeshLab is a tool oriented towards processing large meshes and provides a set of tools for editing, cleaning, and converting meshes. This is useful for producing all-quad meshes for example.