Saturn
Galileo Galilei discovered the first galactic rings around Saturn in 1610. Until 1977 these were the only confirmed planetary rings in the solar system. This is because of their sheer magnitude. The entire rings' system is comprised of billions of particles ranging is size from dust to some the size of mountains. There are two major gaps in the ring system that are caused by two intact moons --- Encke and Keeler.
Scientists believe that Saturn's rings are pieces of comets, asteroids or shattered moons that broke up before they reached the planet. Each of the rings is named for a letter in the English alphabet based on when they were discovered. The rings, in order from nearest the planet to farthest away are: D, C, B, A, F, G and E.
Uranus
The rings around Uranus were discovered in 1977 making it the second planet in the solar system to have confirmed rings. The first six rings were discovered in 1977 with the remain seven being found during the 1986 Voyager 2 mission. The Hubble telescope discovered two new rings in 2005 to bring the total to 15.
The 13 confirmed rings named from furthest to closest: Zeta (1986U2R), 6, 5, 4, Alpha, Beta, Eta, Gamma, Delta, Lambda, Epsilon, Nu and Mu. The two newest rings have not yet been named.
An interesting side note to this discussion is that the rings may have been discovered in 1797 by astronomer Sir William Herschel. Notes in Herschel's hand stated "February 22, 1789: A ring was suspected". This, coupled with a diagram on the facing page, make a strong argument.
Neptune
It was speculated in the mid 1980s that there was a ring around the planet Neptune tentatively named the "Adams ring." This ring was confirmed in 1989 by Voyager 2. The ring was actually three arcs that form an incomplete ring due to the gravitational pull of Neptune's moon Galatea.
There are now five scientifically accepted rings for Neptune; the Adams Ring, the Leverrier Ring, the Galle Ring, the Lassell Ring --- formerly considered a part of the Leverrier Ring --- and the Arago Ring.
Jupiter
Jupiter has three rings, the first of which was discovered by Voyager 1 in 1979. This "Halo" ring is a broad band of ice particles that circles closest to the planet reaching down to the cloud tips. The largest ring, the "Main" ring covers the orbits of two of Jupiter's moons --- Adrastea and Metis --- which are believed to be the source of the dust that makes up the ring itself. The "Gossamer" ring is the outermost of the three and encompasses the orbit of the moon Amalthea. This ring is made up of tiny particles that make it resemble cigarette smoke.