Coins (コイン) are virtual currency used by Ameba for their games. Players purchase coins with real money in order to buy in-game items or roll in certain gachas.
※Coins do not sync from the app to the smartphone/browser version.
※This is how to purchase coins in the app.
Called ボイフレコイン in the app.
※I have iOS. If it works differently for Android, then I can't help you, sorry!
For Boyfriend (Kari), coins are used to buy in-game items and roll in the premium gacha. Ultimately, these are the only uses for coins and you can enjoy and play the game perfectly fine without them. However, if you want higher rarity cards or specific cards, coins are basically the easiest (and sometimes only) way to get them.
You can purchase coins in the app by tapping on the first icon on the navigation bar.
Like all JP apps, you'll have to buy an JP iTunes gift card in order to be able to purchase anything.
Occasionally, there will be deals and discounts in which you can get x% more coins with the same prices. Keep an eye out for ads if you're the type to spend money on mobages. This will typically happen when new/old URs are introduced in the premium gacha.
You can also purchase coins from the Ameba website. The first option is to pay by credit card (クレジットカード). I'm not sure whether a foreign card will work though.