This is a very subjective question, leaving people bound to claim their own country a having the best army in the world. In reality it comes down to what you define as best.
If you define best as "which individual Army is most likely to win a war against a foreign power?", then the answer is undoubtedly the United States. Although they have a smaller Army than the Chinese in terms of manpower, their technology is, at this time, far superior making them more likely to succeed in all out war.
That said, war is rarely straight forward and allegiances are often formed to bring down a common enemy who could not be defeated by an individual nation. For example, if the United States entered into war with A member of the European Union, the other members would probably come to it's aid. In this situation, the eventual victor of the conflict would probably be the allied Europeans as together they have similar technology to the United States but a greater number of troops and a stronger economy.
If the playing field was level in terms of man power and technology the situation changes again, with the emphasis being on the skill of the Individual soldier and the strategy of the army. Here the British probably excel, having regiments like the SAS, Royal Marines and Parachute regiment with a reputation for stringent discipline and intensive training (the Royal Marines have the longest basic training course of any normal infantry soldiers in the world at 33 weeks). Of course the US is not far behind with Delta Force or the Navy SEALs but the size of their special forces makes it harder to train all of the soldiers to the same level that other nations may do. The Gurkhas should probably receive and honourable mention - they fight as part of the British army but hail from Nepal, so in a way they could be seen as an unofficial Nepalese army - in which case they would be the best trained and most skilled of any Army in the world, although lacking technology and manpower, they have proved themselves time and again in conflict.
Historically, Britain and America have always been very strong militarily, although Britain has a richer history. Since the official creation of both the British Army, no foreign power has ever managed to invade Britain (although this could also be down to the fact it i an Island nation). The same can be said for the US; once the British left they have had no need to actually defend their country itself - just allies or overseas territories. Compare these two to places that may have faced similar threats militarily, like Russia, France and Germany and their long standing dominance is very impressive.
Going back even further, the title of best Army in the world could be given to the Roman Army, as for several hundreds of years, they were head and shoulders above their nearest rivals. The Greeks might have a claim to having the toughest soldiers in the Spartans, who even today are world renowned as being seriously good fighters (they won one of most one sided battles in History at Thermoplyae in 480BC. Their entire lives were devoted to fighting and training so it's easy to imagine how they were such good fighters.
In conclusion:
The Top ten "best" Armies at this time in terms of expenditure and man power, training and equipment.
1. United States
2. China
3. United Kingdom
4. France
5. Russia
6. Japan
7. Germany
8. Italy
9. India
10.Saudi Arabia
Historically (looking at the most militarily powerful nations of the last few hundred years) the list should probably go;
1. Britain
2. France
3. Germany
4. Russia
5. USA