Groups work just like tags. You can add any contact to any number of groups and remove them just as quickly. Only administrators have permission to manage groups. Your contacts themselves can never see or manage which groups they are part of.
Once a group has been created you can go to individual contact records and edit the groups they are part of.
Groups, therefore, work differently from segments. Segments include contacts based on a set of conditions. Contacts will be added to or removed from segments automatically based on the segment's conditions.
To view a list of the contacts in a given group you can go to the table of either People or Organisations and choose the group from the drop-down filter at the top of the table.
You cannot directly send a broadcast to a list of people in a given group but you can specify inclusion in a group as a condition within a segment. So if, for example, you manually created a group and you wanted to send a broadcast to everyone in that group you could do so by creating a segment that targets that group as the main condition.