For a body to be classified as a planet it must have a few physical characteristics:
It must have enough mass to have a strong enough gravity to overcome electrostatic forces to bring it to a state of hydrostatic equilibrium.
Hydrostatic equilibrium is important because early in a planets life it is nearly entirely fluid, crust and all
The life cycle of a planet essentially leads to denser heavier metals being at the centre of a planet, surrounded by a mantle which must at some point have been fluid (it can still be solid and called a planet as long as it used to be fluid.)
This is usually driven by its mass but a planet should have an atmosphere. This means it should be massive enough to have a strong enough gravity to hold some gasses to its surface.
More massive planets are capable of keeping lighter gasses, such as hydrogen, bound to them too. I.e Jupiter.
A magnetosphere suggests that the body is still geologically active. This means they have flows of elements that conduct electricity in their interiors.
As specified by the IAU:
A celestial body that (a) is in orbit around the Sun, (b) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, and (c) has cleared the neighbourhood around its orbit.
Minor planets are things such as asteroids, they are usually small mass, rocky bodies. This is different from small mass bodies that might be predominantly ice or water.
Size and Mass
The IAU have not specified an upper or lower limit for mass to be considered a Dwarf Planet, therefore it is predominantly determined by one other feature called:
Orbital dominance is achieved when a body has cleared it's orbit of all other bodies.
For example planets are able to remove small bodies out of their area through impact, capture, or gravitational disturbance.
Any body that is incapable of doing so is therefore classified as a dwarf planet. So if Jupiter was incapable of clearing its neighbourhood of bodies then it too would be a Dwarf planet.