Download the core

Download the recommended release from here: https://www.drupal.org/project/drupal

Extract that packet under your document root and rename it to something like mysite.

My document root for MAMP is /Users/samuli/Sites and the actual site goes here: /Users/samuli/Sites/tutorials/d8/mysite

So now you have this kind of file structure in the mysite folder:

Create database

Go to http://localhost/phpmyadmin.

Click Databases tab:

Write database name, select collation as utf8_general__ci and click Create:

Go through the installation

Visit the site url to start the installation: http://localhost/tutorials/d8/mysite/

Choose language:

Select installation profile:

You can create your own profiles to modify the installation process (like have some modules to be enabled / disabled by default). Bare installation profiles are stored in the profiles directory before you start the installation. Bigger modifications that include core and all other necessary files are called distributions (like Commerce Kickstart or Open Atrium).

Depending on your system/setup, you might or might not see some notices/errors in the Requirements review:

Drupal 8 requires PHP 5.5.9 or higher. Check out other Drupal 8 system requirements here: https://www.drupal.org/requirements

Type in the database name and credentials (for MAMP the default username/password is root). If you didn’t create the database before, the installer will try to create it at this point.

Configure site:

Check for updates automatically will enable the Update Manager module that checks updates and enables you to install/update modules with web interface. You might want to disable the email notifications.

Make sure to subscribe to the security announcements here: https://www.drupal.org/security.

Congratulations, you just installed Drupal 8!