Planet X is a hypothetical planet beyond Pluto. Its existence was argued for on the basis of apparent discrepancies in the orbit of Neptune. This led to the discovery for Pluto, but Pluto's mass was not sufficent to explain Neptune's orbit, so the search continued. However these apparent discrepancies were resolved when the Voyager space probes discovered that Neptune's mass had been badly calculated; with Neptune's newly discovered mass taken into account, there was no longer a need for any new planet to explain Neptune's orbit.
Our most powerful detection techniques are capable of detecting an Earth sized planet 70AU from the Sun, a Uranus sized planet 90AU away and a Jupiter sized planet up to 120AU away (neglecting it's gravitational effects on the Sun). Of course, the sky is very big and the most powerful telescopes can only look at a very tiny fraction of it at a time. Pluto, for comparison, is around 45AU away at the moment.
If any tenth planet exists then it is unlikely to be native to the solar system, comprehensive surveys of the ecliptic have been undertaken, concluding that no planet of Earth size or greater exists in the ecliptic plane closer than 60AU. If it does exist, it will have to be in a highly inclined orbit and so likely to be a captured object and not one that was formed with the solar system.