What is a simple example to explain what a foreign key is?

In a database a primary key is a single value that is unique to each row in your table. For example:

id   | name    | whatever
-------------------------
1      Alice     ....
2      Bob       ....
45     Eve       ....
988    ....      ....

What is a simple example to explain what a foreign key is?

A foreign key is a field that points to a primary key of another table.

Example:

Table Name - Users

UserID    UserName    UserRoleID
1         JohnD       1
2         CourtneyC   1
3         Benjamin    2

Table Name - UserRoles

UserRoleID    Desc
1             Admin
2             Moderator

You can see that Users.UserRoleID is a foreign key which points to the primary key UserRoles.UserRoleID

The use of foreign keys makes setting up relationships on other tables simple, allowing you to link together the data of multiple tables in a nice way:

Example:

SELECT
    a.UserID, 
    a.UserName, 
    b.Desc as [UserRole]
FROM 
    Users a INNER JOIN 
        UserRoles b ON a.UserRoleID = b.UserRoleID

Output would then be:

UserID    UserName    User Role
1         JohnD       Admin
2         CourneyC    Admin
3         Benjamin    Moderator