The title pretty much says it all.
I don't really want the weather info on my home screen or my lock screen. I actually have windows in my house.
Can I remove the weather section from the widget?
This depends on what variant of Android you are running. Stock Android does not include a weather widget but alternatives such as TouchWiz often do. Settings to remove the weather information from the widget may be present depending on your Android variant. Alternatively, you could remove the widget from your homescreen and use another clock widget from the Play Store, such as Digital Clock Widget.
Some widgets only show settings when you create them (as far as I have been able to find - very annoying when you try and customize what they look like). So try deleting it and recreating it.