Office 365 groups is one of a number of Office 365 services that acts as the glue between the various different services you receive as part of your subscription. An Office 365 group not only contains the membership list for a group of people, but also manages the provisioning and access to multiple services like Exchange and SharePoint.
Fundamentally this means when you create an Office 365 group for something like a project, or perhaps a team, they will receive a SharePoint Site, Group Inbox, Calendar, Planner, OneNote and more.
Groups is also the foundation that makes new services like Microsoft Teams possible. Teams is Microsoft’s team chat and collaboration app. It allows you to have long running, fast paced chats with colleagues, voice and video calls, bring files and documents in to work on together and use tabs showing other relevant team information. It uses Office 365 Groups as the foundation for each Team within Teams, using not only the membership list, but also all the underlying group-enabled services for storage of data.
Read my article Set Office 365 group limits to avoid administrative hassles to learn Why a strategy for Office 365 groups is crucial and follow my checklist for best practice Office 365 groups configuration.