Will every object in this Universe attract each other?

Yes. Every object in this universe attracts another object as per the Newtons' law of gravitation.  If there are two bodies (even as big as planets) with masses M1 kg and M2 kg and they are at distance of d from each other,  There is always attraction only.

Gravitational Attraction force between them is = G   M1  M2  / d²

There can be attraction between one body and another if they have magnetic property - north pole and south pole.  But north pole and north pole of magnets repel each other.

There is an attractive force between positive electric charge and  negative  electric charge that is given by coulomb's law  . If they have same sign , then they repel each other.

Moon is attracted to Earth, Earth is attracted to Sun and so on due to gravitational force.

Explanation:

Every Object In The Universe Attract Every Other Object

when you drop and object it falls towards the Earth , this means that the Earth attracts the various objects towards its Centre. Newton said that it is not only the Earth which attracts the other objects . In fact,  every object attract each and  every objects . According to  Newton every object in the universe attract every other objects with a certain  force with which two objects attract each other is called gravitational force.  The gravitational force between two objects acts even if the two objects are not connected by any means .

If the masses of the objects are small then the gravitational force between them is very small .

If one of the objects is very big then the gravitational force  will  be very large .

The universal law of gravitation was given by  Newton. So , it is also known as Newton's law of gravitation .

According to the universal law of gravitation  :

Every body in the universe attract every other body  with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them .