As far as I can tell, the only way to use the "group" feature is in Gmail itself, but not on the Droid's contact list. There may be an app for it, but if so I haven't heard what it is.
Once you are in Gmail (the only way I do this is from a regular computer), click on "Contacts" and then you will see a list of groups and all your contacts. You can make a new group by clicking on the little icon with two heads in the upper left of the screen. Then click on one of your contacts, and the groups that the contact belong to will show up on the right. You can edit those groups from there.
On my Droid, the groups don't seem to show up at all, which is a bummer. -BB