By Archimedes' principle, the upwards force on an object immersed in a fluidis equal to the weight of fluid displaced by it. This is in turn equal to the density of the fluid multiplied by (or integrated over) the volume of the object.
There are three cases that need to be considered: the object is denser than the fluid; the object and fluid are equally dense; or the fluid is denser than the object.
If the object is denser, then the force exerted by the fluid is always less than the object's weight, so the object sinks. This is because the weight of the object is $\rho_o V g$ where $\rho_o$ is the object's density and the weight of the fluid displaced is $\rho_f V g$.
If the object have the same density then the forces acting on the object will balance when the weight of fluid displaced is exactly equal to the weight of the object. This means the object floats when it is fully submerged.
If the object is less dense than the liquid then it floats but is in equilibrium when only partially submerged. This is because the weight of the object is independent of the volume submerged but the weight of fluid displaced does depend on it.
A floating object seems to have zero weight because, by definition, the resultant vertical force acting on it is zero.