Legacy Coins are the premium currency in Legacy of Heroes, and are required to obtain most of the top-end content available in the game. They are usually obtained by paying real money (although the exact procedure depends on the site you're playing LoH on; on Kongregate, for instance, you obtain them by paying Kreds, which are in turn obtained from real money), although there do exist ways to get small numbers of Legacy Coins for free.
Legacy Coins are required to purchase the "advanced" packs in the shop (Prestige Packs and Cosmic Packs), as well as promotional packs, cards, equipment, lunchboxes, and backpacks, and time-limited packs such as the Tempus Packs. Legacy Coins can also be used to purchase Draft Packs (although these can also be obtained by forging busts), as well as purchase special services, such as an instant refill of energy/bonus, a skill point reset, or additional skill points. Legacy Coins can also be exchanged for Bucks at a rate of 1 to 100, but this is a very suboptimal use. (Everything which can be purchased with Bucks can also be purchased with Legacy Coins instead, but this is also not recommended.)
Before the energy system was eliminated in April 2013, a Legacy Coin was awarded to a player for every other level they gained, and the Daily Rewards included Phaeton Coins, which could be forged into Legacy Coins. When the energy system was eliminated, the Legacy Coin for leveling up was also eliminated, but the Daily Reward was improved from the Phaeton Coins to a direct reward of 4-5 Legacy Coins. Legacy Coins can also be obtained as a reward from opening most lunchboxes and backpacks, although these require Legacy Coins to buy in the first place.