Standard practice is to use use most of the time, require occasionally, and do rarely.
do 'file' will execute file as a Perl script. It’s almost like calling eval on the contents of the file; if you do the same file multiple times (e.g. in a loop) it will be parsed and evaluated each time which is unlikely to be what you want. The difference between do and eval is that do can’t see lexical variables in the enclosing scope, which makes it safer. do is occasionally useful for simple tasks like processing a configuration file that’s written in the form of Perl code.
require 'file' is like do 'file' except that it will only parse any particular file one time and will raise an exception if something goes wrong. (e.g. the file can’t be found, it contains a syntax error, etc.) The automatic error checking makes it a good replacement for do 'file' but it’s still only suited for the same simple uses.
The do 'file' and require 'file' forms are carryovers from days long past when the *.pl file extension meant “Perl Library.” The modern way of reusing code in Perl is to organize it into modules. Calling something a “module” instead of a “library” is just semantics, but the words mean distinctly different things in Perl culture. A library is just a collection of subroutines; a module provides a namespace, making it far more suitable for reuse.
use Module is the normal way of using code from a module. Note that Module is the package name as a bareword and not a quoted string containing a file name. Perl handles the translation from a package name to a file name for you. use statements happen at compile time and throw an exception if they fail. This means that if a module your code depends on isn’t available or fails to load the error will be apparent immediately. Additionally, use automatically calls the import() method of the module if it has one which can save you a little typing.
require Module is like use Module except that it happens at runtime and does not automatically call the module’s import() method. Normally you want to use use to fail early and predictably, but sometimes require is better. For example, require can be used to delay the loading of large modules which are only occasionally required or to make a module optional. (i.e. use the module if it’s available but fall back on something else or reduce functionality if it isn’t.)
Strictly speaking, the only difference between require Module and require 'file' is that the first form triggers the automatic translation from a package name like Foo::Bar to a file name like Foo/Bar.pm while the latter form expects a filename to start with. By convention, though, the first form is used for loading modules while the second form is used for loading libraries.