Dragon Power is the money in Mother 3. It is called DP in the game, and is introduced in Chapter 4.
 Gaining DP
DP is relatively easy to gain. Simply defeat an enemy, and you can get a certain amount of DP, depending on the enemy. DP is used to buy everything from stores. Frogs can store DP gained from fights. When an enemy is defeated, the DP gained from that enemy goes to the frogs, to be withdrawn at a later time.
If you are holding DP when you lose in a fight, you will lose it all. What it looks like exactly, is unknown. Is it made of metal coins? Golden pig noses? All we know is it's made of something round, because of the shape of Butch's bag.
 Money and Acorns
According to the secret memo menu, acorns were used as currency before Fassad's arrival but since we never used acorns as money in Chapter 1-3 it seems to have been edited out of the game before release.