When I create a new ticket I take an error message.

[Deleted user]

When I create a new ticket I take an bellow error message "Assignee: The default assignee does NOT have ASSIGNABLE permission OR Unassigned issues are turned off."

[Mathis Hellensberg]

Hmm it doesn't seem like you are using components here, otherwise I would check the components default assignee.

Do you have "Unassigned Issues" turned on? Check in System > General Configuration

EDIT: Try checking if you have a default assignee set for the project under Projects > View all projects > triple dots > project settings.

[John Funk]

Hi Suleyman - Welcome to the Atlassian Community!

Check the Permission Scheme for the project and make sure that the Assignable User permissions apply for the user you are trying to assign on this project. (If you want a User automatically assigned). 

Otherwise, remove the setting where the user is automatically assigned following @Mathis Hellensberg 's recommendations.