How to grant Database level permissions to user in SQL Server

To grant Database level permissions to user in SQL Server (in this example the 'Execute' permission):
In SQL Server Management Studio:
Right-click on a database -> choose Properties -> Permissions:

grant database permissions in ssms

Choose a user to which you are granting the permission, check the 'Grant' (or 'With Grant') box next to the permission, and click OK.

To grant "EXECUTE" rights to user "LGN_test" you can issue the following command:

USE AdventureWorks;
GRANT EXECUTE TO LGN_test;
GO

To grant "EXECUTE" rights to user "LGN_test" with the right to grant "EXECUTE" to other users:

USE AdventureWorks;
GRANT EXECUTE TO LGN_test WITH GRANT OPTION;
GO