There are legal processes in every state (see links below) regarding the position and construction of boundary fences. State legislation and local planning laws cover all aspects of fencing and boundaries, so before you get into an argument with your neighbour about fences it is best to inform yourself about the rules to avoid problems.
There are a lot of urban myths regarding the construction of fences and they cause no end of confusion and aggravation - it seems that the minute you decide to build a fence every man and his dog become experts in what can and can't be built and who pays - 99% of it is absolute rubbish and/or dates back to laws that predate federation. A common myth is (and I have stated this elsewhere on this site) 'if I build a fence [place made up distance here] inside my boundary, I can build it out of [place made up materials here] and it can be [place made up height here] meters high if I want to, and you can't stop me!' WRONG ? Our NFH actually did this, out of spite they built* a 50 meter long second hand tin (tin they scrounged from a junk heap) fence that was over 4 meters high in some places, two meters inside their boundary, it would have looked out of place in a third world slum and yes the council made them pull it down. The funny (dysfunctional) thing was that they had previously fought us for over a year - and lost - to construct a simple wire boundary fence (we are rural residential) on the basis that "we don't want a fence, we dont need a fence" which was, it turns out, NFH speak for "we don't want to pay for our share of a fence".
NSW boundary fences are covered by the Dividing Fences Act of 1991, it covers all the legal requirements regarding the construction of boundary fences and who is responsible for costs, what type, access to property etc. The other consideration is your local council, the planning regulations will have specific guidelines for things like height and construction requirments, so check with them to get information regarding the processes you need to follow.
In NSW (and check in other states because it is probably similar) whatever you do DO NOT go ahead and build or replace a boundary fence without having a signed agreement with the owner of the adjoining property, if you do they will probably have no obligation to contribute to the cost and could possibly obtain permission to have the structure removed and replaced, with you footing the bill for the removal and half the cost of the replacement.
In NSW if the neighbour refuses to contribute or you are unable to agree about the fence and when mediation fails you can make applications for 'Fencing Orders' at the Local Court or the Consumer and Commercial Division of the NSW Civil and Administrative Tribunal (NCAT)
In Queensland you can have fencing diputes handeled by QCAT (Queensland Civil and Administrative Tribunal)
In Victoria contact the Dispute Settlement Centre of Victoria - 1300 372 888
* the Germans have a word for this, it is called a Neidbau which is: A building constructed with the sole purpose of inconveniencing or annoying a neighbour in some way.
Victorian Building Regulations 2006