I have 3 WordPress sites. I want to be able to add posts to one of the sites, and show some of the posts on the other two based on their categories.
E.g. site one has all posts, site two shows posts from site one with category 'site2'
Easiest thing to do is use RSS. See FeedWordPress | simple and flexible Atom/RSS syndication for WordPress and Developer's Guide - Google AJAX Feed API - Google Code and Magpie RSS - PHP RSS Parser for different ways to implement RSS feeds as links or import posts themselves.