Without knowing what you plan to do with your site, it's a bit hard to give a definitive answer. Let's say it's an XBox games site and you have a couple of thousand members. each member is sure to have a favourite game or two and they will want to meet others with a similar interest.
You can set up groups for specific games and your members can choose to join any group that interests them. In the real world, you will allow members to create their own groups and the person who starts a group is the group owner. it's up to him or her to monitor the group's activities and ensure that the group's rules are being honoured. If it's a busy group, the member may invite other trusted members to help moderate the group.
The site owner can control who creates groups or he or she can create groups for other members to use. If the owner doesn't keep an eye on things, hundreds of clowns may start similar groups to the point of stupidity.
I'm not a games person, but let's say there's a game called Mayday and another called Pan Pan. Members could form a group for each of the games, but some dork will come along and start a group called Mayday for left-handed players.
Another example of groups would be a sporting club site. If it's a footy code, you can have groups for Under 18, A grade and B Grade etc.
If you haven't already noticed, Groups are a little like a Forum but there are in fact vast differences. The best thing to do is set up a couple of groups and play with the various options.