What does "balanced triodes" refer to as related to vacuum tubes?

In case this is what you mean, certain tubes are known as triodes. They have 3 elements. 

New Year's assignment-learn about tubes

taste_of_tubes.pdf (ken-gilbert.com)

Yes, most small audio tubes like 12AX7, 12AU7, 12AT7, 6SN7, 6CG7, 6922, 6H30 etc actually have TWO triodes in each glass tube. That's why you sometimes see gear with only ONE tube (one triode handles Left channel and the other Right) or gear with an odd number of tubes like 3. You definitely want balanced triodes, because a big imbalance, greater than say 25% differential, can cause either a left-right channel imbalance or increased distortion, depending on how the tube is used in the circuit. If you have a "matched pair" of these tubes, then technically that's 4 triodes which all should be closely matched!

Whenever I hear a not-so-great sounding tube and then test it (or vice versa), it's quite often the case that some of the triodes are severely mismatched. I use a Maxi-Preamp tester and it can't catch all the bad tubes (e.g. some old 6SN7 have a heater that only goes out intermittently in use) but it catches a LOT. I definitely feel best when all my triodes are matched within 10%. Tube sellers like Upscale and Brent Jesse have the best track record for selling me matched & balanced tubes that are ACTUALLY tightly matched when I test them. Ebay is more of a crap shoot, even if the seller claims perfectly matched & balanced from whatever tester they used. 

Spot on Mulveling....   I will now pay a few extra bucks for matched triodes with dual triode small signal tubes.    They are almost always more uniform,  and test better than just buying un-matched/ un-graded tubes.     

I bought a Hickok tester in August and it was an eye opener.   Tubes from trusted sources were as advertised.   Ones bought in a pinch locally turned out all over the map.    Especially any bought from guitar shops.  Since getting a tester I am pretty confident that music stores get every one else's junk.

Thanks to all for the informative responses.

Thanks, @mulveling  , that was particularly helpful. 

wouldn't it be great if audiogon had 'stickies' for basic knowledge such as this...?

@mulveling +2

@mulveling - Very well said!