To install QUANTUM ESPRESSO from source, you need first of all a minimal Unix environment, that is: a command shell (e.g., bash, sh) and utilities make, awk, sed. For MS-Windows, see Sec.2.7.7.
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 compilers, compliant with C89 and F2003 standards1. 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. If however your compiler is older than 5 years or so, it is unlikely to work. 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).