First we'll explain the ideal type of computing environment, and then we'll say what is the bare minimum you need to run Kaldi. The ideal computing environment is a cluster of Linux machines (any major distribution) running Sun GridEngine (SGE), with access to shared directories via NFS or some similar network filesystem. In the ideal case, some computers on the grid will have NVidia GPUs which you can use for neural net training, and you can reserve these on the queue by adding some extra option to qsub. See Parallelization in Kaldi for more information.
We have started a separate project called Kluster that shows you how to create such a cluster on Amazon's EC2; MIT's StarCluster is a larger and better-supported project that provides the same functionality. Most of the scripts should be suitable for a locally hosted cluster based on Debian or Red Hat; you can investigate Rocks which aims to help you set up a cluster like that.
The bare minimum computing environment to run Kaldi is any Unix-like environment; and it's possible to run it on a single machine, although of course it will be slower, and you may have to reduce the number of jobs used in some of the example scripts to avoid exhausting your machine's memory.
Kaldi is best tested on Debian and Red Hat Linux, but will run on any Linux distribution, or on Cygwin or Mac OsX. We are working on FreeBSD installation scripts.
Kaldi's scripts have been written in such a way that if you replace SGE with a similar mechanism with different syntax (such as Tork), it should be relatively easy to get it to work; we also provide a "dumb" replacement that you can use when there is no queueing system (search for run.pl and ssh.pl in the scripts).
In the past Kaldi has been compiled on Windows; however, the example scripts will not work there, and we are not very actively maintaining the Windows compatibility of the code or the Windows build scripts (we fix problems when we are told about them though).
The following is a non-exhaustive list of some of the packages you need in order to install Kaldi. The full list is not important since the installation scripts will tell you what you are missing.
It can also be helpful if you have an ATLAS linear-algebra package installed on your system. Most systems already have this (You can also search the packages in linux for installation by simple commands like "yum search atlas" or "apt-cache search libatlas"); the best approach is to ignore this requirement for now and see if you have problems when you install Kaldi.
The following tools and libraries come with installation scripts in the tools/ directory so you won't have to install them yourself (note: this is a non-exhaustive list).