To install QUANTUM ESPRESSO from source, you need first of all a minimal Unix environment: basically, a command shell (e.g., bash or tcsh) and the utilities make, awk, sed. MS-Windows users need to have Cygwin (a UNIX environment which runs under Windows) installed: see http://www.cygwin.com/. Note that the scripts contained in the distribution assume that the local language is set to the standard, i.e. "C"; other settings may break them. Use export LC_ALL=C (sh/bash) or setenv LC_ALL C (csh/tcsh) to prevent any problem when running scripts (including installation scripts).
Second, you need C and Fortran-90 (or later) compilers. For parallel execution, you will also need MPI libraries and a parallel (i.e. MPI-aware) compiler. For massively parallel machines, or for simple multicore parallelization, an OpenMP-aware compiler and libraries are also required.
As a rule, QUANTUM ESPRESSO tries to keep compatibility with older compilers, avoiding nonstandard extensions and newer features that are not widespread or stabilized. No warranty, however, if your compiler is older than say 5 years or so. The same applies to mathematical and MPI libraries.
Big machines with specialized hardware (e.g. IBM SP, CRAY, etc) typically have a Fortran compiler with MPI and OpenMP libraries bundled with the software. Workstations or ``commodity'' machines, using PC hardware, may or may not have the needed software. If not, you need either to buy a commercial product (e.g Intel, NAG, PGI) or to use an open-source compiler like gfortran from the gcc distribution. Some commercial compilers (e.g. PGI) may be available free of charge under some conditions (e.g. academic or personal usage).