What is the difference between a Bachelor of Arts Honours and a regular Bachelor of Arts?
Speaking from my experience in Australia -- but I think it holds for most Commonwealth countries.
A Bachelor of Arts (or Science) is 3 year degree, purely by coursework. Though potentially some coursework units may have large research components. You can get into it based on results from high school or some other tertiary entrance exam.
The adding of Honour means the the addition of an extra year, after completion, where you do some coursework, but the focus is on research, you produce somehting which might be called a Honours thesis or similar. Depending on the university, students doing honours may count as post-graduate students, or not (or somewhere between).
It may make sense to think of the Honour year as being a separate qualification -- it is often possible (if unusual) to get into Honours at a different university to where you completed your bachelors.
To get into this Honours pseduo-course, you need to have done well on your units in your bachelors. At the completion your honours results are graded and you get back a result like "1st Class", "2nd Class Division A", "Second Class Division B"; There are more classes, but I've not heard anyone say they have them (presumably because it doesn't sound good to say I have 3rd Class Honours, and/or they don't stick around in academia.).
Honours 1st class is considered roughly equivalent to having done a Masters, as far as getting in to a PhD goes. Thus the Honours pseudocourse can be used as a "shortcut" by high achieving students to get into a PhD a year earlier than they would if they did a Masters -- at the cost of a fair bit of stress usually, because a lot of Honours projects are as hard as Masters projects, and they tend to have a lot of coursework at the same time.
Away from BA and BS, other undergrad degrees may have "built-in" honours. Bachelors of Engineering tend to have them (Particularly if the B.E. would make you a accredited engineer, without having to go on and do Masters). In these cases Everyone does a research based Bachelors Thesis, but only those who have it graded above a certain cut off, and who meet overall coursework grade cutoffs get awarded a Honours classification, and get Honours on their certificate.
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I'm applying to Oxford at the moment for graduate study. Under the section asking what undergraduate degree I'll be receiving, it gives me the choice of "Bachelor of Arts (non-UK)" and "Bachelor of Arts Honours (non-UK)." What is the difference between these two? I might receive departmental honors when I graduate, but I suspect this isn't part of the distinction.