Effective simply means that a method or solution performed very well. Ex. "As a race horse, she is very effective!"
Efficient is a way of saying that it performs well in a cost to gain way. Most commonly you see this applied to money and is describing something as "cost efficient" meaning that you get a large amount of something at relatively low cost (this is often defined by the standard at the time). It can also be applied to other concepts like time,the environment, or energy.
"She is very good at managing time." "She is very time efficient."
"The product costs little to make and sells very well." "The product is very cost efficient."
"These new light bulbs use little energy and have significantly lowered our electricity bill." "These new light bulbs are very energy efficient."