The cart will not show the weight of the items on checkout unless you instruct it to do so. There's a setting to do this in your administration area.
To show the weight of the items on checkout:
In the menu, choose Configuration > Store Settings> General Settings.
You will now see a screen called "Settings". Click on the "Options" tab. Scroll down to the "Checkout" section and choose YES for the setting "Display Weight on Cart Page".
To hide the weight of the items on checkout:
In the menu, choose Configuration > Store Settings > General Settings
You will see the screen called "Settings". Click on the "Options" tab. Scroll down to the section "Checkout" and choose NO for the setting "Display Weight on Cart Page".
