The NFH situation does not usually happen over night, it almost always begins with some form of transgression by you or your neighbour, the loud noise, the barking dog, the fence and who pays what, the tree that clogs the gutters etc. etc. Before you do anything STOP, ASSESS THE SITUATION, think about what you can say and do, as best you can, inform yourself of the facts, try to view things from the neighbours perspective, then take the appropriate action.
Your first and best avenue of approach is obviously to attempt to talk things through with your neighbour in a calm and reasoned way, explain what the problem is, be friendly and try to keep things light hearted, the worst thing you can do is charge next door making ultimatums, attempt to treat your neighbour as you would like to be treated yourself i.e. polite and respectful. Listen to what they have to say, because it may be that they have a situation in their life that has inadvertently involved you e.g. the dogs been barking all night because they have been having to dash off at all hours to look after their sick mother that sort of thing - have a look at these 12 steps from the conflict resolution network.
It is unlikely that you will get a good response from your neighbour if you approach them while you are angry, if you are visibly angry they will probably be automatically defensive so you will be off to a bad start and it will be down hill from there, calm down and be nice (respectful). It is best not to confront your neighbour (or anybody really) with demands of what you want - you should be prepared to make reasonable concessions, try to remain flexible e.g. 'could you please consider shutting the windows when you do drumming practice and may be do it during the day instead of 10.30 at night, is better than 'you can't play the drums ever again because I don’t like it'. If you have arrived at this web site because you have already demanded that your neighbour comply with your wishes then think about perhaps going and apologising for being inflexible and seeing if you can sort things out, it is better to eat a bit of humble pie now than to have years of aggravation and bad feelings. If you want to be pig headed belligerent and vengeful about it then you might consider that it is you who is the NFH not them.
Try not to 'go off half cocked' do some homework regarding the facts surrounding you situation, a common problem in the neighbour feud is the lack of accurate information and the sprouting of urban myths and hear-say about the law, things like fences i.e. 'if I build a fence half a meter inside my boundary I can build it 10 meters high out of old car bodies if I want to and you cant stop me' WRONG or 'I can make as much noise as I like until midnight and you can't do anything about it' WRONG. There are laws, local, state and federal that specifically deal with these things, we no longer live in the 1800's and the majority of legislation that exists in western democracies is to do with law regarding property so it is highly likely that whatever your problem is there is some piece of legislation or set of rules that covers it.