An insignia is a badge you wear to show your position or affiliation. If you're an Eagle Scout, you can wear that insignia, or, if you're a Red Sox fan, you might wear the team logo as your insignia.
The word insignia includes the word "sign" — an insignia is a sign that shows your connection to some organization. Military officers wear insignias that indicate their rank. When you watch the president of the United States give a speech, you'll see on the lectern the official presidential insignia — the circular seal with the eagle inside.