This is your assignment, so i'm not going to write any code for you, but give you some giveaway tips.
Unless you're going to be storing a lot inside the DB, there's probably not much of a use to have a DB just to store it there, and besides it's just a mini project, so keep it simple.
In the end you would probably select Access as the backend, otherwise you might choose MySQL or SQL Server, which you may require the user to install the desktop engine.
Anyways, i suggest that you simply use an XML document to store a collection of user profiles.
What you would do is have a class called User, and have the necessary properties like Name, Password, etc...
Make sure you mark this class as Serializable by applying the Serializable attribute, i.e. <Seriablizable()> Public Class User..... End Class
And all you do is store a List<User> of User objects.
To load and save the user objects you would simply use the XmlSerializer class and invoke the Serialize and Deserialize methods to save and load respectively.
Basically everytime your Sub Main or your main forms InitializeComponent is called you can stick the code which Deserializes the XML document and get your List of Users back, and have that as a global Object.
Then you can instantiate and display the Login form, and implement the Login Button, which just does a look up of the username in the List of User objects, and checks the password.
It's not rocket science, pretty simple.
Hope that helps.