There are three major airports. The largest and the one that handles most international flights is JFK (For John F Kennedy). This airport is located in Queens and is about a 30 minute drive from New York city centre (though it can often take an hour due to the heavy traffic). A cab from JFK should cost $45.

The other airport located in Queens is La Guardia, which handles mainly domestic flights. For international flights apart from JFK, the third main option is Newark. This is situated some way from New York City, in fact it is actually in New Jersey, but it is less crowded than JFK and has good transport links to New York, so it is also a good choice for international travellers.

In fact there are many more airports in tNew York, but these are mainly small and for domestic use between specific destinations only. You can find a full list here.