Yet another English euphemism for the anus (how many do they need?), Roger's Profanisaurus defines the brown eye (sometimes spelled browneye) thus:
The ring (qv) through which the brown
trout (qv) jump on their journey down the pan.
There isn't much explanation needed, really - the anus looks like a scrunched-up brown eye. There's a lot of humour to be found (if you're English) in song lyrics such as "Don't it make my brown eye(s) blue", and in cocktails such as the Brown Eye Opener:
Instructions: Mix and chill, like an Antarctic metaphor
Derived terms also exist, notably brown eye pie (shit). If anyone is interested in reading some really bad historical drama, you can go here: http://home.sprintmail.com/~kenglade/potp-3.htm. Why? Because it features a wise old Native American called...well, you work it out.
Brown Eye Pie
Brown Eye Opener