Telegram Bot HandbookWhat Telegram Supergroups Are
A long time ago, Telegram developers introduced two types of groups: regular ones and supergroups. Supergroups had public links, larger limit for the number of members, and other features designed for big communities.
Apparently, this idea has later been considered rather confusing. Telegram app interface no longer mentions "supergroups" and all groups look the same there, although on the technical level two types remain distinguish. In fact, most of the groups you participate in are probably supergroups.
The main trait of supergroups is that API considers them a special case of channels.
I use the term "groups" both for regular groups and supergroups in the handbook. I've explained developing group bots on the page about groups.
Creating supergroup
A regular group turns into a supergroup while certain settings are changed. As technically the group is replaced with a supegroup-channel, chat ID changes. A supergroup cannot become a regular group again.
Message and group IDs
Message IDs work differently in regular groups and supergroups. In addition, in Bot API regular group IDs start with - prefix while supergroup IDs start with -100.
Gigagroups (broadcast groups)
A maximal number of supergroup members is 200 000. When it's close to the limit, Telegram app suggests admins turn the supergroup into the gigagroup. Gigagroups may contain an unlimited number of members, but only admins can send messages there.
I have no idea why they exist.