If you mean doubling all of the side lengths, then the area would be increased 4 times. Imagine a rectangle that's 2x1. Area is 2. Then, double both side lengths and you get 4x2. The area then would be 8.
If you mean doubling only 1 parallel set of side lengths, the area would be increased 2 times. Imagine a rectangle that's 2x1. The area is 2. Then, double one side length and you get 4x1. The area then would be 4.
Hope this helped!