The Star Rank, or Prf (short for "Preference") Rank, is used to denote a weapon that can only be used by certain characters (usually Lords) in the weapon rank system. For a list of these weapons see List of Star Rank weapons.
Usually, these weapons will have many or infinite uses as there is often only one copy of them per game. However, star-ranked staves still typically have few uses before breaking. Also, they might say that they have a required weapon rank, but it is usually E (the lowest rank possible) and doesn't make any impact on the usage availability. However, certain Prf weapons have a high weapon rank and can be used by multiple people (particularly the Aum Staff in Fire Emblem Shadow Dragon).
In TearRing Saga: Berwick Saga Star Rank weapons work slightly differently - the weapon can be used by anyone with high enough level, but its "chosen" user will be able to use the weapon regardless of level. Weapons that lack visible rank are considered Rank 50 and as such are exclusive to their chosen wielder.