When you click Assignment, the program sets Array2 = Array1. This doesn't work for copying only part of the array but it is fast and easy.

When you click MemCopy, the program uses the RtlMoveMemory API function.

' Declare the memory copying funciton.
Private Declare Sub CopyMemory Lib "kernel32" Alias _
    "RtlMoveMemory" (ByVal Destination As Long, ByVal _
    Source As Long, ByVal Length As Integer)

Private Array1() As Integer
Private Array2() As Integer
Private NumItems As Integer
Private Bytes As Integer

Private Sub btnBuildArray_Click(ByVal sender As _
    System.Object, ByVal e As System.EventArgs) Handles _
    btnBuildArray.Click
    Me.Cursor = Cursors.WaitCursor
    lblTime.Text = ""
    Application.DoEvents()

    Bytes = CInt(SizeText.Text) * 1024 * 1024
    NumItems = Bytes \ Len(NumItems)
    ReDim Array1(NumItems - 1)
    ReDim Array2(NumItems - 1)

    btnForLoop.Enabled = True
    btnAssignment.Enabled = True
    btnMemCopy.Enabled = True
    Me.Cursor = Cursors.Default
End Sub

' Copy using a For loop.
Private Sub btnForLoop_Click(ByVal sender As System.Object, _
    ByVal e As System.EventArgs) Handles btnForLoop.Click
    Dim start_time As Date
    Dim stop_time As Date
    Dim elapsed_time As TimeSpan

    Me.Cursor = Cursors.WaitCursor
    lblTime.Text = ""
    Application.DoEvents()

    start_time = Now

    For i As Integer = 0 To NumItems - 1
        Array2(i) = Array1(i)
    Next i

    stop_time = Now
    elapsed_time = stop_time.Subtract(start_time)
    lblTime.Text = _
        elapsed_time.TotalSeconds.ToString("0.0000")

    Me.Cursor = Cursors.Default
End Sub

' Assign one array to the other.
Private Sub btnAssignment_Click(ByVal sender As _
    System.Object, ByVal e As System.EventArgs) Handles _
    btnAssignment.Click
    Dim start_time As Date
    Dim stop_time As Date
    Dim elapsed_time As TimeSpan

    Me.Cursor = Cursors.WaitCursor
    lblTime.Text = ""
    Application.DoEvents()

    start_time = Now
    Array2 = Array1
    stop_time = Now

    elapsed_time = stop_time.Subtract(start_time)
    lblTime.Text = _
        elapsed_time.TotalSeconds.ToString("0.0000")

    Me.Cursor = Cursors.Default
End Sub

' Copy using MemCopy.
Private Sub btnMemCopy_Click(ByVal sender As System.Object, _
    ByVal e As System.EventArgs) Handles btnMemCopy.Click
    Dim start_time As Date
    Dim stop_time As Date
    Dim elapsed_time As TimeSpan

    Me.Cursor = Cursors.WaitCursor
    lblTime.Text = ""
    Application.DoEvents()

    start_time = Now
    CopyMemory(Array2(0), Array1(0), Bytes)
    stop_time = Now

    elapsed_time = stop_time.Subtract(start_time)
    lblTime.Text = _
        elapsed_time.TotalSeconds.ToString("0.0000")

    Me.Cursor = Cursors.Default
End Sub

Because assignment is roughly as fast as RtlMoveMemory and is much easier to understand, you should use it to copy whole arrays. If you need to copy only part of an array, you can use RtlMoveMemory to avoid using a For loop.