Drupal - How do I programmatically install and enable a module?

There's just one step, using module_enable():

$modules = array('module1', 'module2'); // Array of module names
$enable_dependencies = TRUE; // Whether or not to enable dependant modules

module_enable($modules, $enable_dependencies);

Here is how you would do it with a database update using hook_update_N from another enabled module's *.install file. Then you can visit /update.php in the browser or run $ drush updb on the command line to have this code fired.

/**
 * Enable module1 and module2.
 */
function MYMODULE_update_7101() {

  // Array of module names.
  $modules = ['module1', 'module2'];

  // Whether or not to enable dependant modules.
  $enable_dependencies = TRUE;

  module_enable($modules, $enable_dependencies);
}

For drupal 8 you can use the following:

\Drupal::service("module_installer")->install(["my_custom_module"]);

In drush:

drush php-eval '\Drupal::service("module_installer")->install(["my_custom_module"]);'