What does 'make install' do?
Make is a general purpose workflow program, usually used for compilation. But it can be used for anything.
When you do something like "make all", the make program executes a rule named "all" from a file in current directory named "Makefile". This rule usually calls the compiler to compile some source code into binaries.
When you do "make install", the make program takes the binaries from the previous step and copies them into some appropriate locations so that they can be accessed. Unlike on Windows, installation just requires copying some libraries and executables and there is no registry requirement as such. In short, "make install" just copies compiled files into appropriate locations.
make install does whatever the Makefile author wants it to do. Typically, by this point, it is too late to change the install directory, as it is often known earlier, during the build, so help files and configuration files can be referenced with the correct pathnames.
Many projects use the GNU Autotools to try to improve their portability among hardware and operating system differences. (Different Unix variants use slightly different headers for declarations of functions that are slightly off the usual path -- except most programs need one or another of the ones declared in different locations.)
When a project does use the Autotools, the normal mantra to install it is:
./configure
make
make install
The ./configure typically allows you to use a command line option like --prefix /opt/apache or something similar to specify a different pathname. /usr/local/ is a common default prefix. It is far easier for locally built software to live in one place and distribution-provided software to live in the "main directories": /usr/ /bin/, and so on. (Packagers are very careful to never touch files in /usr/local/ -- they know it is exclusively for system administrators.)
Anyway, the ./configure --prefix /path/to/new/prefix will set a variable in the Makefile that is available when compiling the program, modifying the manual pages so they point to the correct locations for files, modifying configuration files, etc. So make will build the software specifically for the install location you want and make install will install it into that location.
Most programs can run even without the final make install step -- just ./program_name will often start them up. This is definitely a per-project thing -- some, like postfix, qmail, etc., are made up of many different moving pieces and rely on them all working together. Others, like ls or su might be self-contained enough to execute fine from the directory they were built in. (This is not often useful -- but sometimes very useful.)
However, not all projects use the Autotools -- they are huge, complicated, and miserable to maintain. Hand-written Makefiles are much simpler to write, and I personally think just distributing a simple Makefile with configuration variables available is a lot easier on developers and users both. (Though the ./configure ; make ; make install mantra is really easy on users when it works.)
make install does nothing less then executing the install function / section in your Makefile