1. After launching the weather app in iOS 7, click on the area circled in red as shown in the first screen shot to bring up the city list view.
2. Then, as shown in the second screen shot, select the city you want to delete.
3. Next, swipe left on the selection, in the direction of by black arrow as shown in the screen shot below, then click on the delete button that appears.
4. That’s it, notice the selected city has now been deleted from the weather app as shown in the last screen shot. Also, to add new locations click on the “+” sign and to change from Celsius to Fahrenheit, click on the “C / F” toggle switch.
UPDATE (Oct 9th 2013): To Change the list order, simply select and hold the desired city in list view and drag it to its new location.