There's a lot of talk about this over on the Commerce/Business board (
http://community.boredofstudies.org/forumdisplay.php?f=462) but in short;
Cadetship (on BoS at least) usually refers to a program whereby an accounting firm will offer a full time position to a HSC-leaver (or uni student) whereby they will work full time for 2 years (doesn't have to be in one block) whilst studying part time (ie. at night)- and then they will study full time for 2 years and receive a grant from the firm so they do not need to find another job. After this they will have completed their degree, and be elegible for post-graduate training (such as the Chartered Accountant- also known as Professional Year) because they have work experience and a degree. In short, the point of this program is to speed up the time it takes for someone to get their CA. The people also get paid a full time salary, plus benefits and so on, as well as their text books and miscellaneous costs for uni (apart from HECS) paid for- however this varies between firms.
... the other cadetship is an army cadetship, but I'm not really sure how that works.
Apprenticeship is pretty much where you work and learn a trade whilst receiving little pay; but in exchange you get first-hand experience which compliments studies usually undertaken at TAFE. This is common in industries such as building and electrician-work (I don't know what the 'general term' for that is?). I also believe the govt. helps fund apprenticeships- but I'm not exactly sure how this works.
Traineeship is a loosely used term and can apply to an Accounting cadet (after all, the people are 'trainees' even though they are full time employees) or they could be beginners in a trade such as hospitality or services. Usually this refers to a part time job (but it can refer to full time too) where the main point of the job is that the person gets to learn about how things work- and usually receives a very low remuneration for this. However the work experience is invaluable later on.
... btw I could be wrong about all this- this is just what I know based on what I've heard from careers advisor and reading various websites. Can anyone clarify the traineeship and apprentice?