Best - very hard to choose. I was lucky enough to start going the match in the mid-80s. This is a personal favourite from when I was a little kid when I went, but I still remember the 3-1 win over Luton on the last day of 86/87, before we had the title presentation at Goodison. Just amazing stuff. In the modern era, 3-0 over Liverpool, and the 1-0 over United in April 2005 was great too.
Worst - very hard to choose for different reasons. 0-7 away vs Arsenal in 2005. 1-6 home vs Arsenal in 2009. 1-5 home vs Norwich in 1993 (first time I thought we might be in relegation trouble that season). Special mention to that one month period in late 2005, when we lost 0-4 away to WBA, 0-4 at home to Bolton, and 0-4 away to Villa (similar to 1993, I was seriously fearing relegation on the way home from Villa). Also memorably bad were Liege away, and Benfica away, though both fell into the category of 'typical Everton ruining a good day out'.