Our data has improved considerably in the last 20 years.
Mercury: no moons
Venus: no moons
Earth: 1 moon
Mars: 2 moons, Phobos, Deimos
Jupiter: 67 moons, some of which do not yet have proper names. The most famous are the Galilean moons, Io, Europa, Ganymede and Callisto.
Saturn: 62 moons, some of which do not yet have proper names. The most famous, by far, is Titan, which is significantly more massive than any other Saturnian moon.
Uranus: 27 moons, all of which are named after characters in the works of Shakespeare and Alexander Pope. Five are well known; Titania, Oberon, Miranda, Ariel, Umbriel
Neptune: 14 moons, the most massive by far being Triton.
While Pluto, Eris, Haumea, Orcus and Quaoar are considered dwarf planets, they nevertheless have moons.
Pluto: 5 moons, Charon, Nix, Hydra, Kerberos, Styx
Eris: 1 moon, Dysnomia
Haumea: 2 moons, Hi'iaka, Namaka
Orcus: 1 moon, Vanth
Quaoar: 1 moon, Weywot