The Dragon Balls are a Namekian creation. As stated by the Elder Namekian in DB 265, they are the 'spheres of hope...the proof of the wisdom and strength of the children of Namek'. There exist three different sets of Dragon Balls. They can grant the power of wishes to whomever gather all seven, but the appearance, size, and the restrictions on the wishes possible differ from set to set.

Listed below are the three sets of Dragon Balls: the Earthen balls, the Namekian balls, and the Black Star balls. It should be noted that the Earthen balls are unique in that most of the wishes we see in Dragon Ball/Z create evil dragons that emerge much later, around the time of Dragon Ball GT!

Earthen Dragon Balls

Seen in:
Dragon Ball, Dragon Ball Z, Dragon Ball GT

Dragon:
Shenlong (Traditional Chinese Dragon)

Creator/Maintainer:
First set created/maintained by God pre-joining of Piccolo (Devil) and God. Second set created/maintained by Dende.

Appearance:
Red stars within orange balls.

Size:
Smallest of all of the balls.

Restrictions for all of Dragon Ball, up to the rejoining of Piccolo (Devil) and God:
You have only one wish. (Stated several times. IE, Shenlong in DB 212)
You cannot wish for more wishes. (Where stated...?)*
You cannot make a wish that exceeds the power of the creator of the Dragon Balls. (Stated by Shenlong in DB 212)
You cannot revive anyone who has been dead for more than a year. (Stated by God in DB 321)
You cannot revive someone twice. (Stated/implied by Goku in DB 198)
You cannot wish back those who have died naturally. (Stated by God in DB 321)
If the creator of the Dragon Balls dies or ceases to exist, then the Dragon Balls themselves cease to exist. (Proven by the death of God/Piccolo in DB 223 and the rejoining of Devil and God in DB 360)
If the creator of the Dragon Balls is revived, then the Dragon Balls will exist once more. (Proven by Piccolo's statement in DB 293)
The maintaining of the Dragon Balls can be passed onto another individual. (Proven by Dende's creation of new Dragon Balls in DB 393)
Each wish made creates negative energy inside the balls. Too many wishes creates an overload of negative energy, which will result in the Dragons themselves emerging. (Stated by Dark Shenlong in DBGT Episode 48)

Restrictions after Dende becomes the new God of Earth:
You can only have two wishes. However, you may use one wish and wait only four months to make the second. (Stated by Yamcha. DB ?)
Other restrictions remain the same, or can be presumed to remain the same.

Wishes used:
When?: After everyone has been captured by Pilaf. Oolong and Pu'ar have escaped.
Wishmaker?: Oolong
Created?: Six-Star Dragon
1. Get the softest pair of panties in the world.

When?: After the defeat of the Red Ribbon and Pilaf's attack.
Wishmaker?: Son Goku
Created?: Two-Star Dragon
1. Revive Bora, Upa's father.

When?: After the death of Kame'Sennin.
Wishmaker?: Piccolo Daimao
Created?: Four-Star Dragon
1. Grant Piccolo Daimao eternal youth.

When?: After Piccolo Daimao's defeat.
Wishmaker?: Son Goku?
1. Revive all those killed by Piccolo Daimao or his children.

When?: The day before the arrival of Vegita and Nappa.
Wishmaker?: Kame'Sennin
Created?: Five-Star Dragon
1. Revive Son Goku.

When?: During Goku and Freeza's final battle.
Wishmaker: Mr. Popo.
Created?: One-Star Dragon
1. Revive all those killed by Freeza or his soldiers.

When?: After Cell's defeat.
Wishmaker: Yamcha (for first). Kuririn (for second).
1. Revive Cell's victims.
2. Remove the explosives from the bodies of the Artificial Humans.

When?: After the Majin Vegita events and the emergence of Buu.
Wishmaker?: Bulma
Created?: Seven-Star Dragon
1. Revive all of the people killed by Majin Vegita in the Tenkai'chi Budokai, save the 'evil ones'.

When?: After the defeat of Majin Buu.
Wishmaker?: Unknown
Created?: Three-Star Dragon
1. Erase everyone (save our central cast)'s memories of the Buu saga events.

When?: After the defeat of One-Star Dragon.
Wishmaker?: Son Goku
1. Revive everyone killed as a result of villains emerging from Hell/the actions of the Evil Dragons.

Namekian Dragon Balls

Seen in:
Dragon Ball Z

Dragon:
Porunga... aka "The God of Dreams" (Larger Chinese Dragon)

Appearance:
Orange balls with red stars?

Size:
Largest of the balls; much bigger than Earth's.

Creator/Maintainer:
Original set since the 'great cataclysm' (referred to by Dende in DB 260) was created/was maintained by the current Great Elder of Namek; Second set created/maintained by Muuri (Proven by DB 327? 328?)*
.
Restrictions up to Buu saga (presumably):
You have only three wishes. (Stated by Porunga in DB 292)
You can only wish back one person per wish. (Stated by Porunga in DB 292)
The wishmaker must speak in Namekian. (Stated by Dende in DB 292)
If the creator of the Dragon Balls dies or ceases to exist, then the Dragon Balls themselves cease to exist. (Proven by the death of the Namekian Guru, aka Saichôrô-sama, and thereafter stated by Dende, in DB 294)
If the creator of the Dragon Balls is revived, then the Dragon Balls will exist once more. (Stated by Kaio-sama in DB 321)
The maintaining of the Dragon Balls can be passed on from one being to another (Proven by Saichôrô-sama's passing the DB maintenance onto Muuri in DB 327? 328?)

Restrictions after Muuri becomes DB maintainer
You can revive however many beings you want. (Stated by Muuri where...?)*
Other restrictions remain the same, or can be presumed to remain the same

Wishes used:
When?: After Ginyu's defeat, and Dende returns from the Elder Namekian.
Wishmaker?: Dende
1. Revive Piccolo.
2. Transport Piccolo to Planet Namek.

When?: After everyone is revived by Mr. Popo's wish to bring back everyone killed by Freeza and Freeza's soldiers..
Wishmaker?: Dende
1. Transport everyone on Planet Namek, except for Son Goku and Freeza, to Planet Earth.

When?: As Goku and Majin Buu are fighting on Kaioshin-kai, and Dende, Kaiobitoshin, and Elder Kaioushin go to Namek.
1. Recreate Planet Earth.
2. Revive all those who were killed by Majin Buu, save the evil ones.
3. Revitalize all of Son Goku's energy.

Blackstar Dragon Balls

Seen in:
Dragon Ball GT

Dragon:
Original Shenlong (Huge Red Chinese Dragon)

Creator/Maintainer:
Originally created by Piccolo before he split into good/evil halves.

Restrictions for the entirety of their existence:
You can make one wish. (Stated by Original Shenlong in DBGT Episode 1)
If you make one wish with these Dragon Balls, the planet that the wish is made on will explode in one year. (Stated by Kaio-sama in DBGT Episode 1)
If the creator of these Dragon Balls dies or ceases to exist, then the Black Star balls will also cease to exist. (Stated by Piccolo in DBGT Episode 39)

Wishes used:
When?: After Pilaf reaches God's Sanctuary.
Wishmaker?: Pilaf
1. Return Son Goku to a child state.

When?: After Bebi gains control of the entire planet, and the BS DBs are within his grasp.
Wishmaker?: Bebi
1. Recreate Planet Plant