After teaching English to Hungarians, in Hungary, for the past eight years, I would like to share my experiences and observations.

FIRST MEETING
Regardless of what your pupil has told you about his/her knowledge, an assessment of his/her level and ability is necessary. There are two ways to do this: First, their speaking ability; ask them to tell you a little about themselves, speaking for at least two minutes. The second task - comprehension - at this point the pupil really reveals their ability as it is one thing to give an often used paragraph than to listen and understand.
Usually I tell the pupil something about myself, keeping it basic and uncomplicated, in Hungarian, and then ask them to repeat it in English. You will be surprised at the difference between their spoken abillity compared with their ability to comprehend. Set their level from the latter test.

LESSONS
Confidence is a priority when learning a language, so try not to be over critical, during the first couple of lessons don't correct every mistake, let some go, log them in your brain for later.
An hour's lesson isn't really long enough to build confidence level, the pupil only begins to warm up after about forty minutes, so you will be just going over old ground, the real learning part comes in the second part of the lesson. Be patient.

BEGINNERS
This is the most difficult level to teach and particularly if the pupil is an adult. Take a text book and work through a chapter, going over the exercises with your pupil, do this in the lesson as adult pupils are notoriously tardy when it comes to doing homework. On that subject, I give out very little homework, the pupils find it sole destroying, they are paying good money and by just turning up to the lesson, expect to learn, automatically. The teacher is there to do the hard work for them! As soon as it is possible, get off the book and practise their small vocabulary (and grammar) knowledge, once the pupil starts to actually speak English they will feel more enthusiastic about proceeding.
Most pupils come to a private teacher having dropped out of a larger class after only three weeks.

INTERMEDIATE
Personally I find this level the most rewarding as the progression can be remarkably fast. Always take a dictionary to the lesson as seeing the new word in print does seem to reinforce the learning. Ask the pupil to write a dictionary of their own, but rather than alphabetical, according to subject matter, so that by the end of the lesson they have a few new words dedicated to the subject of the lesson's discussion. For example, at this level it is a good idea to talk about the pupil's job, or home. Personalising the lesson does seem to make the pupil more attentive and enthusiastic as it is information they do want to communicate.

ADVANCED
This is more testing. The teacher has to do a fair bit of research to find the holes in the pupil's knowledge, their weak points and work on these. Even after years of speaking a language, certain mistakes can remain (like bad driving). Rather than just constantly correct the error, create sentences in which the word they wrongly use is correct, and sentences in which the word they should be using is correct. Only by correct repetition will it finally sink in. These lessons will run in the form of a conversation, so introduce subjects which will not only stretch the pupil's vocabulary and powers of concentration, but interest him/her aswell.

TENSES
Simple conversations around the present tense are not difficult to conjure up, as I already mentioned, a person's job or home; hobbies are also a good subject.
For past tense study I like to use biographies. Subjects have been 'Famous Hungarians in history' and 'People you admire from history'. Print out a biography in English and in Hungarian and ask the pupil to either translate directly (both) out loud to you, or to summarise. Of course this is only for higher intermediate and advanced pupils, but it does work. You can follow the work up with a discussion, try to encourage a natural dialogue between yourself and your pupil, break out of the pupil/teacher mode a little. Remember the pupil's aim is to have an unstilted, flowing conversation, without feeling self-concious.
Future tense is a difficult concept for Hungarians as they don't often use it, though actually to add the word 'will' is not too much of a challenge for anyone, the teacher just has to find a subject of discussion, and present examples. I have used in the past:
'Tell me about your plans for the weekend'
'Who do you think will win the world cup?'
Turn the tables and get your pupil to ask the questions, after all they are at the lesson to wrack their brain.

PRONUNCIATION
Don't pounce on the poor pupil with pronunciation mistakes, over a period of time they will iron out; the last thing you want to do is scare them off. I have found that the pupil will pronounce the word incorrectly and then (by around the third lesson), will repeat the word correctly without any prompting from the teacher, as they do want to get it right. Do allow the pupil to go at their own pace, at least to begin with.

PROBLEM AREAS
There are certain concepts which are notoriously difficult to get right, I can only give an example which Hungarians find difficult, but every language has there own stubborn areas.
The words (and actions) 'see', 'watch' and 'look'. They always end up in the wrong place. Unfortunately with these it is just a question of repetition, examples, correction, explanation.
Try to predict what problems will arise, for myself, as I live in my pupils' country and not vica versa, it hasn't been too hard, but then there are always those odd surprises.