Which Planet In Our Solar System Is The Farthest From Earth?

To answer your question briefly: Neptune is currently considered to be the farthest planet from our own planet, Earth, within the solar system. However, this was not always the case.

Pluto was was known as the ninth planet in the solar system ever since its discovery in 1930. However, in 2006, Pluto was demoted to dwarf planet status, when the International Astronomical Union reclassified the definition of what was considered to be a planet.

This left Neptune as the outer-most planet in the solar system.

Solar System Planets: The Distinction Between Planet and Dwarf Planet
It is currently agreed that there are eight planets in the solar system, and five dwarf planets. By definition, dwarf planets occupy the slightly-vague territory between asteroid and planet.

Finding a solid definition of what a planet can actually be defined as, is difficult. The best definition I've found is that a planet:

• Orbits the sun (or, if it's outside our solar system, then it must orbit another star)
  
• Is not a moon (i.e doesn't orbit a planet itself)
• Is large enough to have pulled itself into a roughly spherical shape by the force of its own gravity, (unlike asteroids, which remain as inert, mis-shapen lumps of rock)
• Is large enough to have cleared a distinct space for itself within its orbit of the sun - it will have done this by pulling in and absorbing nearby material, using the force of its own gravity
  

However, in spite of these definitions, some astronomers still consider both Pluto - and another orbitary body known as Ceres - to fall into the classification planets. Some astronomers have a very conservative attitude to this matter, and believe that changing the classification of Pluto has led to confusion amongst the public.