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I know there are some awesomely far spacecraft that have been travelling for decades away from earth, but which is the farthest? And has it discovered something amazing?

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Do you mean space-craft in general, or satellites? E.g. Voyager is a space-craft. Sputnik was a satellite. –  Everyone Dec 3 '13 at 13:13
    
@Everyone, pretty much. –  Jamd Dec 3 '13 at 20:58

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up vote 20 down vote accepted

Wikipedia states that Voyager 1 is currently farthest from Earth; it gives the distance as of Dec 3 2013 as 126.95 AU ($1.899 \cdot 10^{10}\text{ km}$) from Earth. At that distance, the speed of light delay is approximately 17 hours 36 minutes. By Dec 7 2013 it was listed as at a distance of 127.03 AU from Earth, which means it is moving away from Earth at a speed of 0.02 AU per day, corresponding to a speed (relative to Earth) of roughly 35 km/s. The speed relative to the sun is stated as approximately 17 km/s. The probe was launched, along with its sister Voyager 2, in August 1977; Wikipedia states that Voyager 2's current speed relative to the sun is slightly above 15.4 km/s at a distance from Earth of 103.00 AU as of Dec 7 2013. Since you're asking about the farthest, I focus on Voyager 1.

Exactly what might count as "amazing discoveries" is of course a matter of personal opinion rather than a strict scientific assessment, but Voyager 1 does seem to have contributed to our understanding of the outer limits of the heliosphere. As TildalWave pointed out in a comment (since deleted), it has also provided evidence of many other things, including that it has:

The feat of communicating with the spacecraft (also here) might actually be quite significant in and of itself; Voyager 1 communicates with Earth (through the Deep Space Network) on frequencies near 2.3 GHz and 8.4 GHz. Unless I'm getting the calculations wrong (although the numbers do appear to be within the ballpark), free space propagation losses alone is in the neighborhood of 300 dB on 2.3 GHz and 320 dB on 8.4 GHz. I don't know what sort of antenna gains the DSN sports, but even assuming a -100 dBm receiver sensitivity (including antenna gains on the spacecraft) and 100 dBi gain on Earth, you'd still need to pour +100 dBm into the 2.3 GHz feed antenna to make it. +100 dBm is 10 MW of power.

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+1 for pure awesomeness. –  Undo Dec 7 '13 at 17:19
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@Undo Those distance figures really make you question whether that commute to work is so bad after all... :) –  Michael Kjörling Dec 7 '13 at 17:20

As of 7 February:

Voyager 1 Spacecraft, ephemeris for Fri 7 February 2014, 02:13 UTC Right Ascension: 17h 11m 54s Declination: +12° 01’ 12” (J2000) [HMS|00:00:00|Dec] Distance from Sun: 18,968.76 Million Km Distance from Earth: 19,024.36 Million Km Magnitude: N.A. Constellation: Oph

given an AU as 149.597 million km... 127.1 AU and climbing. Note that distance will vary by ±2 AU over the year, and is travelling about 1AU per 100.5 days, or about 3.6 AU per year.

http://theskylive.com/ephemerides-computation?obj=voyager1#lat|77.9799442515|lon|102.02461516200002|fov|80

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3.6 AU/year puts the speed at ~17.12 km/s (both relative to the sun), which closely matches the 17 km/s relative to the sun I stated in my answer. –  Michael Kjörling Feb 7 at 12:22

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