How far apart do settlements have to be in Catan?
At least two edges away from every other building, yours or anyone else's. In practice, none of the three intersections next to your new settlement can have a settlement or city.
The rule
This is the distance rule. A new settlement must be at least two edges from all other buildings, whoever owns them.
After setup, a new settlement must also connect to at least one of your roads. During setup, your second settlement doesn’t need to connect to your first.
Cities don’t need a new spot. You upgrade one of your existing settlements.
Example
Ana wants to build on the intersection right next to her own settlement. She can't. She has to extend her road one more edge and build on the next intersection.
Why players argue about it
Players assume the rule only applies to opponents' buildings, or only during setup. It applies to every building, all game.