Hanoi: Airport transport

Many hotels offer free taxi services from the airport when you arrive. Check with them before your arrival and they will meet you with your name written on a piece of paper. Just get into the car and enjoy the ride (takes about 45-60 minutes into town, depending on whether the traffic is heavy, very heavy or extremely heavy.

Even if they don't offer you a free pickup, they are often very competatively priced.  Allowing them to arrange it for US$15-20 is money well spent.

The commonest: Taxi

When exiting Noi Bai International Airport, visitors will see lines to queue for taxis, with numerous signs giving the fixed price: VND 350,000 ($16) for a sedan (toll charges are included in the price) and VND 380,000 for a MPV (7 seater, Toyota Innova).

There are a few companies operating this route below:   Visitors will pay directly to the driver when arriving at hotel. No toll fee or any other charge.  The taxi usually has the meter running, but you should pay the fixed fee. 

Note: it is a good habit to write down the number plate of the taxi in case the driver want to cheat you.

By Minibus
Continue to the right past the taxis and you will find the minibuses:
Airport Minibus one way trip costs VND40,000 ($2). Minibuses leave the airport when they are full.
You can also hire a whole minibus for VND500,000. 
They will manage to squeeze you and your luggage in somehow, but prepare to be squashed.

Public Bus
Keep going past the minibuses and through the car park and you will find the public buses:
Bus #7 (Cost VND6,000) connects the airport and Cau Giay bus station (in the West of the city).
Bus #17 (Cost VND5,000) will get you to the Long Bien bus station (near the Old Quarter).
It can be crowded and you will need to transfer from bus station to your hotel.  They may not let you on if you have lots of luggage.