Consider these sentences:
Die Katze sitzt auf meinem Schreibtisch.
Wir wohnen auf dem Lande.
Sie fahren auf das Land.
For the first case auf is translated as on, in the second as "in", the third as "to". I figured out these translations from google translate.
What is it that controls the meaning of auf in a sentence?
To expand a bit on the answer given by infinitezero, many common words, especially prepositions, are impossible to translate directly without taking a number of factors into account. Instead, it's best to learn the possible meanings of a word and under what circumstances it can be used. These additional factors are usually referred to as "context". It's easy to miss the fact that word usage is so complicated in your own language because you learned this information a very young age and use it without thinking about it. But when you learn a new language you have to learn a new set of meanings and circumstances for each word. This is not easy, and I hope no one gave you the impression that learning German would be as simple as memorizing the word in German for each word in English. You can get a hint of how complex English prepositions are by looking them up in a dictionary, for example Wiktionary gives 37 meanings for "on" as a preposition; this does not include meanings as other parts of speech and its use in prepositional verbs. German is no less complicated.
To add an additional wrinkle in German, prepositions, such as auf, which describe a location have different meanings depending on the case of the noun that follows. I think it's best to think of auf as two related words; an accusative version and a dative version. This is similar to the way you might use "on" or "onto" in English depending on the circumstances.
The German auf is used (with a dative noun) when something is on top of something else, but in a number of other circumstances as well. In this sense the best translation is usually "on", but this is not always the case and "on" has additional meanings that don't translate to German as auf. For example in English when you say "I'm sitting in the chair" you mean you're sitting on top of the chair, and in German you'd say Ich sitze auf dem Stuhl. If you say "The picture is on the wall" you don't mean the picture is on top of the wall, but that it's attached to the wall, and German uses a different preposition, an, for this meaning: Das Bild ist an der Wand.
But auf as other meanings. It's used when something appears as an image in something else: Ich bin auf dem Bild. – "I'm in the picture." As a location it's used with large open areas as in your second example: Wir wohnen auf dem Lande. – "We live in the country." (As infinitezero mentioned, Land(e) is considered a large open area in German. I gather that the dative Lande is uncommon in modern German, so it would be more usual to say Wir wohnen auf dem Land.) Another example: Die Kinder spielen auf der Straße. – "The children are playing in the street." Large institutional buildings fall under this heading as well: Sie arbeitet auf dem Flughafen. – "She's working at the airport." As with English prepositions, the meaning of auf can be figurative: Ich bin auf einer Reise. – "I'm on a trip."
Your third example uses the accusative version of auf, which is used when the location is used as a destination or direction of movement, hence its use with fahren. In the main meaning of "on top of" this might be translated as "onto" rather than "on": Die Katze springt auf das Bett. – "The cat is jumping onto the bed." But in your example the meaning is as a large open area and "onto" is not appropriate in this case. English tends to use "to" in a variety of circumstances to indicate a destination, which is how you get "to the country" from auf das Land. In this case the meaning of auf is more specific than "to" since it tells you what sort of destination it is, not just that you're going somewhere.
Nothing. Prepositions differ from language to language.
English counter example:
Wir gehen über die Straße.
Wir fahren über Berlin.
Wir fliegen über Berlin.
We walk across the street.
We drive via Berlin.
We fly over Berlin.
In German, Land is regarded as a large open place, that's why auf is needed in both latter cases.
