Read it Through
After you have written your essay, it is important to eliminate any errors of grammar or spelling, and to make sure that all your points are clearly phrased and make sense. I know that it is very boring to read through an essay when you have just spent hours working on it and all you want to do is go to bed/ to the pub/ to have breakfast, etc. You needn't read through it absolutely immediately, but you should find fifteen minutes or so at some point before you hand it in to check through it. This is especially important in exams (see below).
It is a good idea to think about what you hope to get out of your supervisions, both generally and in particular instances. Spend some time preparing for supervisions by thinking about which aspects of the topic gave you trouble or particularly interested you, and what questions would be the most useful ones to ask your supervisor. The purpose of the supervision is to have a two-way discussion about your essay. The one thing it should not become is a general lecture on the topic by the supervisor. Make sure that you get your say, and that you take a hand in guiding the discussion in the ways that would be most helpful for you.
Take notes of the important parts of what you and your supervisor say during your supervisions. Notes from supervisions provide a record of the way that your thoughts developed during conversation with your supervisor — and of any new facts or ideas with which your supervisor furnished you — which will be invaluable when it comes to revising for your exams.
Some people think, because they are quite clear what is being said and that they have understood it at the time of the supervision, that there is no need to make notes for future reference — they have truly taken the ideas on board through talking and listening during the supervision. These people are mistaken and will kick themselves when it comes to revising and they find that they cannot remember what that brilliant insight was the one that hit them during the supervision — the one that was so lucid and penetrating that there was no way they could possibly ever forget it ... until six months and three sets of supervisions on totally different papers later...
Of course, you do not need to write down every word that you and your supervisor say — use your discretion about what to write down, and when. For example, if you are having a particularly animated and engaging conversation, it's probably best not to keep interrupting the flow of it to write things down. But you could, perhaps, at a convenient pause in the conversation, ask your supervisor if she could give you a couple of minutes to note down what you have just been talking about. Another possibility is making some notes immediately after the supervision, while the issues you covered are still fresh in your mind.