A public Team is a group that anyone can search for and join. They will also have a public Message Board forum.
For a private Team, you must have a coded url or a Team invitation sent to you to join. You can enter the coded url into the address bar at the top of the page or click on the link in the Team invitation in your SparkMail, and it will take you to the Team page to join. Private Teams also have a private Message Board forum that can only be seen and used by their group members.