How do I get my site to show up in the search engines?
Google Search Console (aka Webmaster Tools) is where you tell Google about your website and let them know how to index your site in the search engines. Google will eventually find your site and index it on it's own but this tool can speed up the process as well as help to inform google about what to index on your site.
Think of Search Console as the application for getting your site in the Google Search Index so that visitors can find you. Then the Google Analytics application is where you see who visited your site. Each tool has it's own features and benefits but they also have the ability to link both applications together to a web property. Since the Search console knows about what keywords people used to search your site this information can be presented to Google Analytics and also shown in your data results.
Connect Google Analytics with Webmaster Tools
Generate a Sitemap and submit it to Google
Configure Wordpress to build and submit a sitemap
Configure Drupal to build and submit a sitemap
Bing (Microsoft)