R: Computing language for statistical computing and graphics
RStudio: An integrated development environment (IDE) for R
BUGS: Software implementing MCMC sampling
JAGS: Software implementing MCMC sampling; similar to BUGS
Once you have installed R, you can install additional R packages from within R using install.packages. In particular, you will need to run install.packages("R2OpenBUGS") and install.packages("R2jags") before using those packages.
As a syntax reference for BUGS/JAGS, the appendix of our textbook is quite helpful. Online references include the OpenBUGS manual and the JAGS manual.