For power projection, you need aircraft carriers. The United States Navy has the best carrier force in the world and by a very wide margin. France is the only other country to operate a CVN, and the UKs carrier force is the second-largest in the world (three ships total). The American force brings a lot of capability to the mix twelve CVNs and the resulting air wings (anywhere from 40-50 fighters, each). However, the Royal Navy is no slouch the liberation of the Falkland Islands in 1982 was probably the most impressive power projection in the last 25 years.
For amphibious operations, again, the United States has the best capability. China, France, and the UK also have significant amphibious forces as well. This is another means of power projection, because it provides a means for a country to carry out a raid or to deliver ground troops to a place without port facilities (usually for a forcible entry). The best force for an amphibious operation, though, is the royal Navy. Again, the Falklands War is what puts them over the top. Since 1950, the United States Navy and Marine Corps have not really had to conduct a major amphibious operation against a defended beach. Desert Storm, Iraqi Freedom, and Vietnam were mostly fought from ground bases. The United Kingdom, however, had to carry out just that sort of operation in 1982 and to do it half a world away. While the United States has learned lessons from that war, the British did it for real and that counts a lot.
When it comes to ship-to-ship combat, the best is pretty easy to determine in blue-water combat (operations far from shore, where the only real support you will get will be an occasional maritime patrol aircraft) it is the United States Navy with its carrier battle groups. India and Pakistan have fought several wars, and sheer quantity will place the Chinese high on the list as well. Japan also is high on the list with a large force of destroyers equipped with surface-to-surface missiles as well.
For operations close to shore, this changes. The Israelis have probably the best force in the world in terms of capability and there is the past experience of wars with Arab States (where the Saar-class missile boats were in combat with Arab Komar and Osa-class missile boats). Norway and Sweden have forces that are geared towards fighting in littoral (coastal) waters Norway was, in fact, planning to face off against the Soviet Union. That said, they have had no combat experience. Indonesia and Singapore also have superb littoral warfare capabilities a necessity in both countries cases. Singapore is a small country, Indonesia has a large number of islands. The Israelis are probably still the best, though, since they have maintained a high level of readiness.
Now, we come to ASW (Anti-Submarine Warfare) the hunting of submarines. The Americans have a large force of submarines and maritime patrol aircraft. Also, many of the American surface vessels can carry helicopters for anti-submarine work, so they are a contender for the top role. Taiwan has a number of older naval vessels (their best ASW vessels are ex-U.S. Navy Knox-class frigates), but they train hard because of a large force of Chinese diesel-electric submarines. Japan and the UK also are island nations who have had to deal with massive submarine campaigns against their navy and merchant marine the latter did so twice. The British are probably the best, due to past experience and the fact their equipment (sonar, helicopters, and torpedoes) is as good as the Americans.
Submarine warfare is much the same only it involves using submarines. The United States, Russia, the United Kingdom, China, Japan, and India all have large submarine forces, while Australia has a small (six diesel-electric submarines) but very well-trained force (just ask any U.S. Navy units that have gone on exercises against the Australians). The largest and most technologically advanced is the United States Navys although the British submarine captains are said to be probably the best in the world. In this area of naval warfare, it is a tie between the U.S. and the U.K. The Americans have the larger force. That said, the British have a very well-trained force that has sunk a ship in combat and won a naval war. The torpedoes fired by HMS Conqueror in 1982 were the decisive shots of the Falklands War the sinking of the General Belgrano scared the Argentinean Navy back into port, and cleared the way for the liberation of the islands.
Ultimately, the best navy at a given task will be the one that has trained the most for that task and which has quality personnel and equipment that is up-to-date and well-maintained. Overall, the best navy is the one that can do the most thing pretty well (even if they are not the best). The United States Navy does just that, and many of the navies that are somewhat better in other areas belong to its allies. Harold C. Hutchison (email@example.com)
Navies have many tasks to operate far from home ("project power"), to support amphibious operations, battles with other navies (both high seas combat and coastal operations), anti-submarine warfare, maritime interdiction and protecting convoys. So, which Navy does what the best?