Most important thing to mention regarding installing software on Linux is that it's much more reliable and easy to install software from your distribution (this is it's purpose!). Only use make install
if there's no other way (consider alternate programs as well).
Common mistake of Windows users is to download programs from different places and try to install them, forgetting to check out their distribution packages, which could be installed with a single click or command (in the package manager).
Also remember that one of the main reason for absence of virus infections on Linux is that software is (or should be) installed from central (trusted) location instead of the many random sites.
As an additional note, while distributions like Ubuntu always contain outdated packages, there are also distributions like Arch Linux that are always up-to-date (though they don't offer literally every program ever released for Linux, like Debian/Ubuntu).
As for what exactly would happen when you use package manager, well it would check dependencies, download packages, unpack, put all files to their appropriate directories, according to FHS and distribution's own guidelines, and some other routines which you could probably find out about from the man page of the package manager.
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My answer to http://stackoverflow.com/questions/528399/what-should-linux-unix-make-install-consist-of gives a brief overview of the usual kinds of things done by
– dmckee --- ex-moderator kitten – 2011-11-27T20:33:23.467make install
steps from a programmers point of view.