1. Drupal core search
Drupal core provides a search module by default.
What you get ? Out of the box you have search pages for nodes and users.
- Search box send to search result
- search results are group per tab User & Content
- You can disable search pages
- you can add custom search pages based on nodes and users
Pros Very easy to configure
Limitations This module is pretty basic and limited. For example no out of the box support for custom entities. If you need a custom entity then you would need to implement a Search plugin for each custom entity.
2. Search API
This module is better for more advanced functionality. It has a good abstraction to allow using different search servers.
What server option to go for?
Starting: If you are just starting the Database Backend is good enough.
Advanced: If you have a much bigger site it is best to go use an external backend like Solr & Elastic search.