What is the busiest airport?
Note: Since this answer was first written, updated statistics have been published. The original answer was written against 2015 figures, with the update from 2017 figures.



There is no one simple measure of "busiest" when it comes to airports, there are usually (at least) three.


Busiest by passenger traffic - defined as total passengers emplaned and deplaned, with transit passengers counted once
Busiest by cargo traffic - loaded and unloaded freight and mail measured in metric tonnes
Busiest by traffic movements - total number of landings and takeoffs of all aircraft




The most recent statistics produced by Airports Council International are from 2017. 

By passenger traffic the top 3 are


2015 


Hartsfield–Jackson Atlanta International Airport,  USA
Beijing Capital International Airport,  China
Dubai International Airport,  UAE

2017


Hartsfield–Jackson Atlanta International Airport,  USA
Beijing Capital International Airport,  China
Dubai International Airport,  UAE



By cargo traffic


2015


Hong Kong International Airport, Hong Kong
Memphis International Airport, USA 
Shanghai Pudong International Airport, China

2017


Hong Kong International Airport, Hong Kong
Memphis International Airport, USA 
Shanghai Pudong International Airport, China



By traffic movements 


2015


Hartsfield-Jackson Atlanta International Airport, USA
O'Hare International Airport, USA
Dallas/Fort Worth International Airport, USA

2017


Hartsfield-Jackson Atlanta International Airport, USA
O'Hare International Airport, USA
Los Angeles International Airport, USA 



(Aside: Wow, those Americans like their traffic movements - 8 of the top 10 are USA! | 2017 update: 7/10)



There are a few non-standard claims too. They are interesting, if not winners.


Memphis International is the Home of FedEx, and as such was the highest freight airport for a number of years.
Gatwick Airport is the busiest single-runway airport (There are actually two runways, but only one is ever in use)
Wittman Airport hosts an aviation event which temporarily raises it to one of  the busiest airport in the world for that short period.
The airports of London (Heathrow, Gatwick, City, Luton, Stanstead &amp; Southend) host the most passengers annually for a single city's airports combined. 

(Don't get me started on Southend being a "London" airport - it's at least 20 miles from what could sensibly be called London)




References:


ACI: busiest airports in 2015
ACI: busiest airports in 2017
Wikipedia: Busiest airports