Have you ever played with a see-saw or a balance board? When you and your friend play on it, you try to balance your weight on both sides to keep it level. If one of you is heavier, then that side will go down, and the other side will go up.
Now, imagine if the see-saw was very, very sensitive, like a scale that can measure even the tiniest weight difference. That is what Eötvös rule does - it measures the weight difference between two objects that are very, very close to each other.
Scientists use Eötvös rule to study how gravity affects objects. They will put two objects next to each other, and then measure how they are affected by gravity. If the objects have different weights, then they will be pulled towards the Earth differently. Eötvös rule can actually measure very tiny differences in weight, so scientists can use it to study things like the Earth's gravitational pull, or the way different materials are affected by it.
So, in short, Eötvös rule helps scientists understand how things are affected by gravity by measuring the weight difference between two objects that are very close to each other.