Taking gravity as an example:
You have this idea that a small mass has less gravity, so it should have a weaker attraction.
So in the example, if X has the small mass, and Y the bigger mass, then you'd think that Y attracts X more strongly than X attracts Y.
This intuition comes from seeing that the bigger mass clearly moves less.
If I drop a spanner (small mass), I experience the spanner falling to the Earth (big mass), not the Earth rising to meet the spanner... so it is tempting to think that the Earth attracts the spanner more than the spanner attracts the Earth.
However - the fact is that if X is attracted to Y, then Y is also attracted to X with equal strength.
X and Y experience the same strength force, in opposite directions. As Nugatory says, it is law #3.
Since F=ma, and the forces are the same, the smaller mass will experience a bigger acceleration, and, so, will have the biggest movement ... giving rise to the intuitive feeling.
There is still a valid idea that big masses have more gravity somehow ... this is taken up by the concept of the gravitational field. To find the force on Y due to X, you take the gravitational field of X, and multiply it by the mass of Y. To find the force on X due to Y, you take the gravitational field of Y and multiply by the mass of X.
So, if X has mass m and Y has mass M, M > m, the gravitational field due to X is ##g_X=Gm/r^2##, and the gravitational field due to Y is ##g_Y = GM/r^2## (notice that ##g_Y > g_X##)... and you can work out the forces yourself to see how they end up the same even though the strength of gravity differs.