There are 2 distinct versions of Microsoft Teams.

Business Microsoft Teams

This is also known as "Teams for Work or School". This is the version of Microsoft Teams every company uses. Meeting links will have the form https://teams.microsoft.com/....

You can sign up for a Business Microsoft Teams account here.

Personal Microsoft Teams

This is also known as "Teams for Home or Small Business". Meeting links will have the form https://teams.live.com/....

If you create a Microsoft Teams meeting with your personal Outlook or Microsoft account, the meeting will be considered a Personal Microsoft Teams meeting.