Introduction
Start simple, with a sentance stating which way you will be arguing, and a second sentance summarising your argument. For example, "I will be arguing that..."
Body
You only have around 500 words and a short period of time, so this has to be fairly brief. Write one or two (short) paragraphs detailing your argument as to why you are correct. Keep this to the point and explain your reasoning - dont waffle!
You should also include one argument on the other side of the debate, or a counter argument, which you should subsequently argue against in order to invalidate it. Again, this shouldn't be overly complicated.
Conclusion
Like your introduction, this should be very brief.Restate your argument, and finish with the wording of the question, e.g. "...therefore, euthanasia should be legalised in the UK."
Other tips
Don't overcomplicate it, and don't try to use specialist language. Your essay is only used to assess how you write an essay and if you are clear and logical, not your content.
You don't need any background knowledge, so when you pick your question try to think which would be easier for you to write this brief essay on - this might not be the one you are an expert on!
Always spend 5 minutes planning. Failing to plan will severly limit how clear your argument is - and this is the whole point of the essay.
Don't stress too much about the essay. It is probably more important to spend your time practising the LNAT comprehension style questions rather than the essays. Try a couple of timed essays to have a feel about it, but don't spend too much time on this section.