If density of water and any object is same, then will that object sink in water? What will happen to it?

Any object, when immersed in a liquid, will displace certain quantity of that liquid. The object will encounter an upward force (also known as buoyant force) equivalent to the weight of the fluid it displace. If the weight of the object is more than this upward force of liquid, it will sink (till it displaces more liquid to support its weight). If, on the other hand, the object has less weight than the displaced liquid, it will float.

In this case, the object has the same density as the water. This object, when immersed in water, will displace a certain amount of water. Since the densities of the object and water are same, the buoyant force will be exactly equal to the weight of the object and this will be a case of neutral buoyancy. The object will neither float nor sink and will be perfectly balanced by the displaced water. In fact, the object will not move unless another force acts on it or it was already moving. 

Hope this helps.