macro that would increment the new value I am assigning in the first column by 1 each time

A quick search on the net gave me this:

Private Sub Worksheet_Change(ByVal Target As Range)
    Dim KeyCells As Range

    ' The variable KeyCells contains the cells that will
    ' cause an alert when they are changed.
    Set KeyCells = Range("B1")
    
    If Not Application.Intersect(KeyCells, Range(Target.Address)) _
           Is Nothing Then
           
        
        Range("A1").Value = Range("A1").Value + 1
        
       
    End If
End Sub

This assumes that A1 wil increase by 1 when B1 changes. You need to open the VBA editor and then select the worksheet that this will be used on and insert the code there.
This should then run in the background and increment A1 each time B1 changes.

In A1, put 1 to start the list.

In A2, put this formula and copy down:

=IF(B2=B1, A1, A1+1)

or:

=A1+(B2<>B1)

Ahhh I've re read your post after seeing JBeau's post and my solution isn't what you need so please ignore this.
JBeau's solution is exactly what you want.

Guys, I want to thank you so much for your help and detailed responses.

I understand JBeau's solution and I think it does work, the only thing I don't get is the "copy down" part. Do I have to put that formula in every cell (A1, A2, A3, and so on)? There is about 1000 rows. I know I can click and drag but is that what must be done? Thank you so much!

I've marked this thread as SOLVED for you.
Next time, select Thread Tools from the links above and mark this thread as SOLVED. Thanks.