(Don't dismiss this as dry science-y stuff, give it a read.)
There is news out there that Voyager One is about to leave the solar system. Here is a pretty good article, with pictures and diagrams. What they mean by 'leave the solar system' is that it is almost beyond the influence of the sun. Specifically, it is beyond the protection of the sun and facing the cosmic energies that exist between the stars.
Voyager One launched in 1977 and is now about 16 light minutes away. That means that it takes light (or a radio signal) a full sixteen minutes to reach us. The probe is one of the fastest man-made objects ever made and it's only 16 light minutes out there. How fast is it going? Let's do some back of the envelope math and see if we can figure it out.
It took 35 years to go 16 light minutes away. After about 50 years it will be one light day (and yes I'm rounding for simplicity.) That means it will take 365 X 50 years (18,250) before it will be one light year away from Earth. The nearest star from our sun is Alpha Centauri which is about four light years away. That means an object moving at Voyager One's speed and going in the right direction would take about 73,000 years to get to our nearest neighboring star. That, to put it mildly, is a long time.
Now 1) Voyager One is not moving in the right direction and 2) it will lose power long before then so this exercise is completely academic. They currently think it will lose power some time in the next ten or fifteen years. Until then they'll keep checking the sensors.
So Voyager One is the furthest man made object away from Earth. How long will it hold that title? Voyager Two is only twelve some light hours from Earth and will never overtake its sister ship. Most recent planetary probes weren't designed to go to interstellar space, instead they have gone into orbit around various planets. NASA's current approach has been to plunge them into the atmosphere of Jupiter or Saturn. (They want the probes destroyed so that they won't clog up the radio bands.)
Right now a probe called New Horizons is heading all the way out to Pluto. After zooming by the former planet, it will continue further out and eventually reach interstellar space. It had a faster launch than Voyager One but it didn't have the gravity assist boosts that the earlier probe did. As a result it is a bit slower and will never be further away.
We can currently build a faster ship and send it out but until (if?) we decide to do so, Voyager One will remain the furthest man made object from Earth.