Power Coin

In the Mighty Morphin Power Rangers series, the Power Coins were magical golden coins which enabled the Power Rangers to morph. Though the series started with only five, several more were revealed later. With the exception of the White Tiger (which was created by Zordon and Alpha 5), possibly the Dragon Coin (which may have been created by Rita), and the cloned Dragon Coin (which was created by The Wizard of Deception) all the Power Coins were created by Ninjor.

Note: may not be canon In an early storybook explaining the Power Ranger's origin, Zordon challenged Rita Repulsa to a coin toss in order to settle a long conflict. The loser would be imprisoned for all time. The six 'Power Coins' were used with Zordon winning five tosses and Rita winning one. This explains how Rita may have come into possession of a Power Coin, and makes the Green Ranger a true Power Ranger.

Original Coins

When Zordon and Alpha 5 summoned the original Power Rangers to the Command Center, they presented the teenagers with the first five Power Coins:

Zack Taylor received the Mastodon Power Coin, enabling him to morph into the Black Power Ranger and control the Mastodon Dinozord. The coin later passed to Adam Park. It was also used by Abraham as Wild West Black Ranger. The coin was last seen in Adam's possession and restored by the Sentinel Knight.
Kimberly Hart received the Pterodactyl Power Coin, enabling her to morph into the Pink Power Ranger and control the Pterodactyl Dinozord.
Billy Cranston received the Triceratops Power Coin, enabling him to morph into the Blue Power Ranger and control the Triceratops Dinozord. It was also used by William as Wild West Blue Ranger.
Trini Kwan received the Saber-Toothed Tiger Power Coin, enabling her to morph into the Yellow Power Ranger and control the Saber-Toothed Tiger Dinozord. The coin later passed to Aisha Campbell. It was also used by Miss Alicia as Wild West Yellow Ranger.
Jason Lee Scott received the Tyrannosaurus Power Coin, enabling him to morph into the Red Power Ranger and control the Tyrannosaurus Dinozord. The coin later passed to Rocky DeSantos. It was also used by Rocko as Wild West Red Ranger. The coin was later shown to be in the possession of its original bearer, Jason, and is fully functional as seen in "Forever Red".

The Power Coins were carried in the Rangers' Power Morphers, which were worn on their belts when morphed. The five coins bore images of their sentient beasts.

When Lord Zedd took over command of the moon palace, the Rangers changed Zords, but kept their original coins. In the beginning of Season three, when the Thunderzords were destroyed by Rito Revolto, the original Power Coins are disconnected from the damaged Morphing Grid in a power surge that damaged the coins, rendering them unstable and essentially useless. Adam was shown to be in possession of the Mastodon Coin, broken down the middle, in the Power Rangers: In Space episode "Always a Chance". When Adam reappeared in 'Once a Ranger', he was again using the Mastodon Coin, but this time it was shown to be completely undamaged. This could be attributed to the Sentinel Knight, who also restored Tori's and Kira's powers, which had been completely lost.

Dragon Coin

Eventually it was revealed that Rita Repulsa possessed a sixth Power Coin, which she gave to Tommy Oliver when she brainwashed him into serving her. Though the spell was later broken, Tommy retained the coin, which allowed him to morph into the Green Power Ranger and control the Dragonzord. The Dragon Coin bore the image of a three-taloned claw.

Not long after Tommy joined the team it was revealed that Rita Repulsa was able to drain the Dragon Coin's power supply via a Green Candle whose wax had covered Tommy while captive in her palace and possessed a link to the Dragon Coin's power retention ability. Initially, the coin was passed to Jason in order to retain what was left of the coin's capabilities, but was returned to Tommy when he rejoined the team. Though Zordon reenergized Tommy's coin several times, its power was eventually drained completely at the hands of Lord Zedd's Turban Shell monster. Zedd tried and failed to harness the remaining green powers to energize his own evil ranger team. The Wizard of Deception would later create a clone of Tommy, imbued with the Green Ranger powers. The current status of the coin that was in possession of the Tommy clone, be it the original or copy, is unknown.

White Tiger Coin

Zordon and Alpha took matters into their own hands and created a new Power Coin, the White Tiger Coin, which they gave to Tommy. This allowed him to morph into the White Power Ranger, the new leader of the Power Rangers team, and control the White Tigerzord. The White Tiger Coin bore an unusual design like the one on his body shield's centerpiece rather than an image of an actual tiger.

The White Tiger Coin was damaged with the original five coins when Rito Revolto destroyed the Thunderzords.

Ninja Coins

In the beginning of Season three, it was revealed that Zordon and Alpha held a map that supposedly led to the mythical Temple of Power and the creator of the Power Coins, Ninjor. After the destruction of the Thunderzords and concurrent damage to their existing coins, the powerless Rangers traveled to the Desert of Despair to locate Ninjor. He gave them the new Ninja Coins and the corresponding power of the Ninja.

Tommy received the Falcon Coin, allowing him to morph into the White Ninja Ranger and the White Power Ranger, and control the Falcon Ninjazord.
Adam received the Frog Coin, allowing him to morph into the Black Ninja Ranger and the Black Power Ranger, and control the Frog Ninjazord.
Kimberly received the Crane Coin, allowing her to morph into the Pink Ninja Ranger and the Pink Power Ranger, and control the Crane Ninjazord. This coin was later passed to Katherine Hillard when Kim decided the leave the group.
Billy received the Wolf Coin, allowing him to morph into the Blue Ninja Ranger and the Blue Power Ranger, and control the Wolf Ninjazord.
Aisha received the Bear Coin, allowing her to morph into the Yellow Ninja Ranger and the Yellow Power Ranger, and control the Bear Ninjazord.
Rocky received the Ape Coin, allowing him to morph into the Red Ninja Ranger and the Red Power Ranger, and control the Ape Ninjazord.

Like the original Power Coins, the Ninja Coins bore images of their namesake animals.

The Crane Coin was stolen by Katherine (under Rita's spell), thus disabling access to the Ninjazords and weakening the Pink Ranger. During this period, the Rangers adopted the Shogunzords instead and kept with them even after the Crane Coin was recovered.

Lord Zedd and Rita Repulsa destroyed the Ninja Coins when an experiment by Billy to return the rangers to normal age backfired. The rangers soon after became the Power Rangers: Zeo by using the Zeo Crystal, but the Ninja Power Coins were lost forever.

From Sentai To Power Rangers Conversion

The Power Morpher and Power coins used in the show are actually the Dino Buckler toys released in Japan for the Super Sentai Series Zyuranger. New Power Ranger labels were created and placed over the original stickers. Tommy's Power Morpher was a Dragon Buckler, also from the same series. Current Power Ranger series continue to use the toys as props.
The original core team's were the only coins accurately released as a toy in North America. The Dragon Coin was released in Japan market (as part of the Zyuranger toy line). The White Tiger coin was redesigned from its symbolic claw to a tiger. In Dairanger, Kibaranger never had a power coin, so the the coin used in the show was custom made from resin (possibly made from combining Kibaranger coin symbols used in Dairanger).
The Ninja Coins received a McDonald's Happy Meal release, however, they were dual-sided (two faces per coin) and did not fit in the morpher toy. However, many fan prop makers have created appropriate replicas.

Current Status

Since the end of Mighty Morphin Power Rangers, only three coins are seen or suggested to remain. The Mastodon coin was shown to be in Adam's possession but greatly damaged. In the Operation Overdrive two-parter "Once a Ranger", it was powered back up to full strength by the Sentinel Knight. Ironically, while the Dragon Coin was the first to lose its power, there is evidence suggesting that it remains one of the only fully functional coins. The last confirmed fully functional coin is the Tyrannosaurus coin, which belongs to Jason. However since Jason has shown to still retain his powers, it can be assumed that Zack and Trini also have retain their own coins, morphers and powers, making the five fully functional coins the Dragon Coin, the original Tyrannosaurus and Saber tooth tiger Coins, and not one but both Mastodon Coins.

Unanswered Questions

While the original five Power Coins and the Ninja Coins were all explicitly stated to have been created by Ninjor, the origin of the White Tiger Coin is a bit murkier. It is possible that, since Zordon and Alpha created the White Ranger, they may have minted his coin as well.
The Dragon Coin's origins were never given in the series. There are obvious similarities that suggest the coin was minted by Ninjor, however, these similarities may be coincidence in a similar nature to other sixth rangers that resemble the core team, but differ in origin (ex. the Gold Ranger). The only definitive evidence given to the coins origins can be found in secondary sources, namely the official Mighty Morphin Power Rangers comic book, and an interview done by Jason David Frank, both of which cite the creator of the coin to be Rita. Though, most fans do not consider these sources canon, and maintain that the coin was created by Ninjor and at some piont may have been stolen by Rita.
In the power transfer of the original Red, Yellow and Black rangers to the new ones, when the original rangers (Jason, Trini and Zack) leave the Command Center for the last time, they remain in ranger form (albeit due to the actors portraying them having quit many episodes earlier). Equally, the new rangers (Rocky, Aisha and Adam) are also in ranger form, but the coins being physically given to the new rangers were not seen. Therefore, there is evidence that the original power coins were copied and were not destroyed, which would explain how Jason had his powers in "Forever Red" and suggests that Zack and Trini are also still in possession of their original powers.

Miscellaneous

It was often stated that only those in possession of a Power Coin (and Alpha 5) could enter the Command Center (this caused havoc in Green with Evil, when the evil Green Ranger was allowed in, since he possessed the Dragon Coin). However, in Mighty Morphin' Power Rangers: The Movie, Ivan Ooze was able to enter the Command Center by oozing through the door. Also, Goldar and Rito Revolto were able to enter the Command Center when they planted the explosives that destroyed it and made way for the "Power Chamber."
In "A Zeo Beginning, Part Two", outtakes at the end show that at the point where Zordon says "Protect them (Zeonisers) as you did your Power Coins", Jason David Frank calls out as a joke, "But we didn't protect our power coins, we lost them!" causing Alpha to short circuit.
In Zyuranger, the Black Ranger's coin was a Mammoth. In Mighty Morphin' it's referred to as a Mastodon.
In the "Master Vile And The Metallic Armor" 3-parter, when the rangers have climbed up the Shogun Megazord to gain access to it they pose in a circle. The camera pans around at each ranger's waist, but they show the original coins and not the ninja coins, which is a mistake.