What is the best known non-governmental organization currently working
exclusively on Israeli-Palestinian peace?

Dear Jonesem, 

I'm afraid I cannot agree with Geof-ga, despite the fact that I
actually once knew a guy who worked as a spokesman for Shalom Achshav.

I think that for several years now, since the ourbreak of the
so-called "Second Intifadha" (in 2000), "Peace Now" is no longer
relevant. There are several other initiaves that are important today,
each reflects some of the confusion of the Israeli "peace camp" since
2000:

"The Geneva Accord"
<http://en.wikipedia.org/wiki/Geneva_Accord> - initiated by Yossi
Beilin, the leader of Israel's left wing party, Meretz; and by
Palestinian counterparts.

"The People's Voice"
<http://en.wikipedia.org/wiki/The_People%27s_Voice> - initiated by Ami
Ayalon (in the beginning a former Navy admiral and a non-governmental
peace activist; nowadays an MK for the Labour Party, Avoda/Meimad) and
the Palestinian politician/activist Sari Nusseibeh.

Both these initiatives are backed by "moderate", zionist, political
and social movements. Another vocal movement (and in my opinion the
most important that doesn't involve any parliamentarians) is "Gush
Shalom" (Peace Bloc):
Gush Shalom
<http://en.wikipedia.org/wiki/Gush_Shalom> 

I hope this answers your question. Please contact me if you need any
clarification on this answer before you rate it.

Probably the Israeli Peace movement, Shalom Achshav (Peace Now),
though its support is currently declining because of the success of
the Hamas Party in the Palestinian election. For more information, see
http://en.wikipedia.org/wiki/Peace_Now

probably red cross.