When the Custom option is selected, the search slider will appear on the right hand side. You can find out more about each individual search option in User Guide - Search for items.
The Cypher search option is only enabled for users with the default SUPER ADMIN and ADMIN roles. If you would like to enable it for other users, either modify an existing role to enable the Cypher Search or create a new role specifically for that - see Admin Center - Roles panel:
You need to enable at a minimum the READ and CREATE permissions to perform a query. The Cypher language is powerful and enabling the UPDATE and DELETE permissions will allow users to perform cypher queries which can update or delete existing application data.
Starting 2.5.x, it is possible to remove objects from a custom view using the right click Remove object(s) option (you can also use the DELETE key):
Starting 2.11.x, it is possible to use the Undo and Redo options in the bottom left corner of the view, i.e. specifically to reinstate an object you have removed.