First, open the Network and Sharing Center by clicking on the network icon in the notification area. Next to the current network name is a large icon, in this case a house representing a Home network (Note that it is the larger icon in the 'View your active networks' section I am referring to - thanks to the commenter who pointed this out).
Click on the house icon and a Set Network Properties dialog box will appear that allows you to rename the current network and optionally change the icon.
But we want to delete other network names, so click on the link at the bottom labelled Merge or Delete Network Locations.
Now you'll see a complete list of all the network locations your PC has connected to. Select those you want to get rid of - you can't delete them all at once, as selecting multiple names activates the Merge option. This lets you combine all the networks into a single name - I'm not too sure what use this is. Note that you can't delete any networks that are currently in use.
Update: As one kind commenter has pointed out below, you can delete multiple names by selecting them with Shift-click or Ctrl-click, then pressing Delete. Just make sure that none of the networks selected has an 'In Use' status, which is what I did - if you do this, the Delete button will be greyed out.