On public variables from the help file:
Public Statement Example
This example uses the Public statement at the module level (General section) of a standard module to explicitly declare variables as public; that is, they are available to all procedures in all modules in all applications unless Option Private Module is in effect.
[VBA]Public Number As Integer ' Public Integer variable.
Public NameArray(1 To 5) As String ' Public array variable.
' Multiple declarations, two Variants and one Integer, all Public.
Public MyVar, YourVar, ThisVar As Integer[/VBA]

to reference a textbox from a standard module you would have to fully qualify it. For example:
[VBA]Sub a()
UserForm1.Show
UserForm1.TextBox1.Text = "Test"
End Sub
[/VBA]