configure deals with differences between the libs and tools installed on the target system - see this page for an overview - and generates a Makefile (usually a Makefile in each sub dir too).
A Makefile is a script usually defining several types of operation - you select the operation by specifying it as an argument to 'make'.
'make clean' (by convention) should removed all the compiled libs and programs from the current dir and sub dirs. This should be redundant where you've just unpacked an installation - but its good practice to make sure the packager didn't leave any stray object files around.
'make' will compile the libs / programs
'make install' usually just copies the relevant files from the working directory tree to their final locations on the target system.
You can postpone 'make install' almost indefinitely - but if your code depends on linking to libs already on your system then the resulting code may break if these are updated in the meantime.