Problem: We would like to create an application page that doesn't require login and any user can open that.

1. Make sure the website that calls the application page is available for anonymous access.
If the URL to the application page is http://server01/_layouts/testpage.aspx, then http://server01/ has to be accessible by anonymous user.

2. When you create an application page in Visual Studio 2010, by default when you open the page in view code, you see the class below:

public partial class testpage : LayoutsPageBase
{
    protected void Page_Load(object sender, EventArgs e)
    {
    }
}

LayoutsPageBase by default fires the login prompt. So just replace it with equivalent class that can be called in an unsecure context. That class is UnsecuredLayoutsPageBase.

So your code should look like this.

public partial class testpage : UnsecuredLayoutsPageBase
{
    protected void Page_Load(object sender, EventArgs e)
    {
    }
}

3. UnsecuredLayoutsPageBase has a read-only property called AllowAnonymousAccess. This property returns false by default which leaves you with an extra step to override it and make it return true. So your final code will look like this:

public partial class testpage : UnsecuredLayoutsPageBase
{
    protected override bool AllowAnonymousAccess
    {
        get
        {
            return true;
        }
    }
    protected void Page_Load(object sender, EventArgs e)
    {
    }
}

Deploy the page and enjoy opening the page without login prompt.

OK, I have done this but it still prompts me for the login info! Well the only other thing that I can think about now is go to the site settings and make sure the master page as a published and approved version.

Note: All the credit of this post goes to Dr. Zewei Song and Kevin Chen. I have just packaged and formatted their work.  

Cheers
Alireza