I manage the New Foundations Home Page , where you should look for enlightenment about this subject! I have just updated it after a hiatus (March 2005)
A new draft paper about defining equality in NF and related systems is here (Postscript) and here (Latex source) .
I have written a book, Elementary Set Theory with a Universal Set , which HAS APPEARED as no. 10 in the Cahiers Series of the Center for Logic in the department of philosophy at the Catholic University of Louvain (but the book is now out of print: see the next paragraph). See here for a local file with order information, or look at the Cahiers site. This book is roughly parallel to Halmos's Naive Set Theory but the development is carried out in NFU (New Foundations with urelements) instead of ZFC. Earlier electronic versions of the book contain various inaccuracies and infelicities; please consider getting the final version! There are of course some residual inaccuracies and infelicities, here humbly noted in an errata list. Here is the Postscript version of a draft of the errata slip.
With the permission of the publishers, the book is now posted online here. This is an ad hoc corrected version; there will be an official second edition online eventually.
Some letters to the FOM list about topics related to NF are accessible from the "top" of my web page.