What cards use coin tokens?
Although the cards using coin tokens are already listed in the answer by RM Bee (including the comments), Coin tokens appear in currently four expansions (Seaside, Prosperity, Guilds, and Renaissance) and are used slightly differently in each set.

Please note that all the named cards have additional abilities besides generating or converting Coin tokens, but as the question is about Coin tokens this answer also focuses on them.



Seaside

The first expansion set to feature Coin tokens is Seaside. Using its card Pirate Ship, Coin tokens can be collected by individual players on their own Pirate Ship mats. Alternatively the collected tokens can be converted to +$ using the same action card, but the Coin tokens are not used up, so they remain on the Pirate Ship mat. 

In later turns, this basically makes Pirate Ship kind of a Treasure which is possibly huge, but needs to be played as an action to generate money.



Prosperity

Prosperity also features Coin tokens with its Trade Route card, but this time the Coin tokens are placed on top of the victory card piles in the supply (one each) during setup. Once a card is gained from such a pile, the token is moved onto the Trade Route mat. 

Amongst other effects, playing Trade Route then generates money according to the number of tokens on the mat. Again, the Coin tokens remain on the mat when used. In contrast to Pirate Ship, however, the Coin tokens on the Trade Route mat do not belong to a single player, but can be used by anyone.



Guilds

Using Candlestick Maker, Baker, Plaza, Merchant Guild, or Butcher from the Guilds expansion, players can get Coin tokens individually. The Second Edition printing of Guilds uses the name "Coffers" for these tokens; when a card says "+1 Coffers", you add a coin token to your Coffers mat. The details or conditions on how to get them are different for each card.

The Coin tokens from these cards can be used like other Treasures during the Buy Phase to pay the costs of the cards. Each coin generates $1 and is returned to the coin supply after using it. So basically, you can use these action cards to save some money and use it in later turns, which you cannot do with regular Treasures, obviously.



Renaissance

The newest expansion, Renaissance, brings back the same coin tokens from Guilds. It includes a "Villagers/Coffers" mat, which can replace the Coffers mat from Guilds. Certain cards give you either Villagers or Coffers, which you track by placing coin tokens on the correct half of the mat.

Coffers work exactly the same as they did in Guilds. Villagers can be spent at any time during your action phase to add +1 Action.

Cards that provide Coffers include Ducat, Silk Merchant, Swashbuckler, Patron, Spices, Villain, Exploration, Pageant, and Guildhall

Card that provide Villagers include Acting Troupe, Recruiter, Sculptor, Lackeys, Patron, Silk Merchant, Academy, and Exploration.