Basically, you need to produce a line of reasoning that supports your point of view. Tutors and lecturers need to see that you can:
In effect this is about finding your academic 'voice' (what you think and argue in relation to your subject).
Being able to critically analyse and evaluate your subject is your starting point here. You can't expect to know what your argument is going to be before you've done your critical investigation (reading, experimenting etc) into the subject. This often involves making judgements about the validity of different authors' positions and the way they uphold them. You need to stand back and scrutinise issues, looking at both sides of a case, strengths and weaknesses, patterns and connections in ideas, evidence, arguments and findings. You will need to analyse and evaluate your own ideas (which will probably be developing and changing as you carry out your research). Your argument emerges from this research and analysis. For help with this visit the Critical thinking and Reading strategies sections of this site.
Methods of investigation depend on the subject and your approach. What's important is that the methods used allow you to establish effective reasons, evidence and examples for your arguments.
Conclusions need to follow logically from your reasoning and evidence. Make sure that the steps to your conclusions are clearly laid out.
You argue in order to persuade and you won't do this if you don't know what you're talking about. Make sure your research is relevant and up to date.
'Talking through' ideas is an invaluable way of working out what you think about them. Try this with peers - or just talk things through out loud! (A dictaphone is useful here.)
You may find it helpful to apply the same criteria used in Evaluating Arguments to assess if your own arguments that you construct in your essays are convincing.