Can a foreign key refer to a primary key in the same table?

I think the question is a bit confusing.

If you mean "can foreign key 'refer' to a primary key in the same table?", the answer is a firm yes as some replied. For example, in an employee table, a row for an employee may have a column for storing manager's employee number where the manager is also an employee and hence will have a row in the table like a row of any other employee.

If you mean "can column(or set of columns) be a primary key as well as a foreign key in the same table?", the answer, in my view, is a no; it seems meaningless. However, the following definition succeeds in SQL Server!

create table t1(c1 int not null primary key foreign key references t1(c1))

But I think it is meaningless to have such a constraint unless somebody comes up with a practical example.

AmanS, in your example d_id in no circumstance can be a primary key in Employee table. A table can have only one primary key. I hope this clears your doubt. d_id is/can be a primary key only in department table.

This may be a good explanation example

CREATE TABLE employees (
id INTEGER NOT NULL PRIMARY KEY,
managerId INTEGER REFERENCES employees(id), 
name VARCHAR(30) NOT NULL
);

INSERT INTO employees(id, managerId, name) VALUES(1, NULL, 'John');
INSERT INTO employees(id, managerId, name) VALUES(2, 1, 'Mike');

-- Explanation: -- In this example. -- John is Mike's manager. Mike does not manage anyone. -- Mike is the only employee who does not manage anyone.

Sure, why not? Let's say you have a Person table, with id, name, age, and parent_id, where parent_id is a foreign key to the same table. You wouldn't need to normalize the Person table to Parent and Child tables, that would be overkill.

Person
| id |  name | age | parent_id |
|----|-------|-----|-----------|
|  1 |   Tom |  50 |      null |
|  2 | Billy |  15 |         1 |

Something like this.

I suppose to maintain consistency, there would need to be at least 1 null value for parent_id, though. The one "alpha male" row.

EDIT: As the comments show, Sam found a good reason not to do this. It seems that in MySQL when you attempt to make edits to the primary key, even if you specify CASCADE ON UPDATE it won’t propagate the edit properly. Although primary keys are (usually) off-limits to editing in production, it is nevertheless a limitation not to be ignored. Thus I change my answer to:- you should probably avoid this practice unless you have pretty tight control over the production system (and can guarantee no one will implement a control that edits the PKs). I haven't tested it outside of MySQL.

Eg: n sub-category level for categories .Below table primary-key id is referred by foreign-key sub_category_id

A good example of using ids of other rows in the same table as foreign keys is nested lists.

Deleting a row that has children (i.e., rows, which refer to parent's id), which also have children (i.e., referencing ids of children) will delete a cascade of rows.

This will save a lot of pain (and a lot of code of what to do with orphans - i.e., rows, that refer to non-existing ids).