kendall wrote:
"So, what do you call it when you play the D in the regular position and your little finger on the first string in the 5th fret?
When you drop the 6th string down to D it makes a rather pleasing sound.Of course, it doesn't fit every time, so it can't replace the D."

Do you mean the D maj chord that looks like this?

__0___
___X_X
____X_

The first string has the F#, which is the third, so you're doubling the fifth and eliminating the third. It has no third, so it's neither major nor minor. I notate it as D with a subscripted 3 that's crossed-out, as above. Some people deny that it's a chord and call it, um, something different. A ditone or something.

This chord (or whatever) is used frequently in Renaissance music, e.g., John Dowland.

I like the sound, too, and use it fairly often.