The report search bar works similarly to the app level search bar. The report level search bar enables users to search through the list report they are on for the record they want. The more accurate your search is, the more specific the result. E.g. simply searching 'd' will return all records beginning with 'd' whereas searching 'demo' will return all reports that has 'demo' in its name. The search bar searches through records fields that have been specifically selected so that if you search for a record that has Demo in the title it won't search for records with Demo in the description unless selected.
The Global search bar is similar to the report search bar but instead of searching one specific report for the records you want (based filtered reports that could limit your search), it will search the whole app and create a app search report based off the layout of the default report and display it for you there. Please remember that this is a system provided report and will not stay there for the next time you come back to the app unless you use the global search bar again.