Privacy

Privacy is a feature that allows you to change who is/isn't allowed on a server, only the owner may do this.

Public

Anyone can join the server, by clicking join random world, searching, or by ip. These servers can be stolen from easier, but can prosper faster, and be a lot more fun to play on than another server.

Searchable

These servers require the player to search it, so it takes either luck or knowing a player who is on the server to enter. Searchable is a great way to allow some friends to get on, but keep others from destroying your world before you get ownership signs.

Private

The most secure servers of them all are private, and only you and other whitelisted players may join. This is even more secure than searchable or having a password. Only mods, admins, and the owner can whitelist someone, so only those you (or another mod/admin) trusts may get on, great for a friends-only server.

Password

Passwords are not officially a level of privacy, but they do a similar thing, filter who can join. These are in the area between searchable and private, anyone who knows the password may enter, making the world extra secure, but regular players can tell friends the password to log on, making private servers more secure.