This answer is meant for users of Unix and Unix-like systems. Users of other operating systems should see See Where can I get Emacs for macOS, MS Windows, etc?.
Most GNU/Linux distributions provide pre-built Emacs packages. If Emacs is not installed already, you can install it by running (as root) a command such as ‘dnf install emacs’ (Red Hat and derivatives; use ‘yum’ in older distributions) or ‘apt-get install emacs’ (Debian and derivatives).
If you want to compile Emacs yourself, read the file INSTALL in the source distribution. In brief:
(Replace ‘VERSION’ with the relevant version number, e.g., ‘28.1’.)
tarprograms, which are standard utilities. If your system does not have them, these can also be downloaded from https://ftp.gnu.org.
tar can uncompress and extract in a single-step:
tar -axvf emacs-VERSION.tar.xz
cd emacs-VERSION ./configure # configure Emacs for your particular system make # use Makefile to build components, then Emacs
make completes successfully, you can go on to install it.
(See What should I do if I have trouble building Emacs?, if you weren’t successful.)
Note that ‘make install’ will overwrite /usr/local/bin/emacs and any Emacs Info files that might be in /usr/local/share/info/.