This is a really tricky question to answer, mostly because it's so broad. Every navy in the world changed significantly over the course of WW2 - the RN of 1939 was very different to the RN of 1945, and the same is true for the USN, IJN, Kriegsmarine, Regia Marina, Soviet Navy or French Navy. In addition, it covers such a wide range of things that can go into a comparison that to give a full answer to the question really requires a book. As I don't really have the time to write one, this comment will cover three main areas: ships (including design, building and weaponry), training and tactics, and carriers and air operations. I'm also going to include the various Dominion navies under the general banner of the Royal Navy. While these forces were nominally independent, they copied British traditions and techniques, and often relied strongly on British facilities for training and equipment.

At the start of the Second World War, the RN was almost a fleet of two different parts. It had a solid core of WW1-era ships, in varying stages of obsolescence. While these ships could be very useful when modernised, as with HMS Warspite, money had been short in the 1920s and 1930s, and many ships still bore the weaknesses of their original design. The second part of the fleet bore the stamp of the various naval conferences and limited naval budgets that had characterised the 20s and 30s. The earliest ships of this part, the Nelson class battleships, and County class cruisers were built to the limits of the Washington Naval Treaty, while the York class were designed to squeeze as many ships from a limited budget and tonnage allowance. Moving into the 1930s, the RN began to construct large 6in cruisers to stick to the letter of the London Treaty, starting with the Leander class, and ending with the various subclasses of the Town group. The King George V class also saw serious limits due to the London Treaty, being armed with 14in guns. When designing ships, the RN tended to design ships that well-fitted its strategic situation and planned tactics, sometimes at the expense of fighting ability. This tended to manifest in shorter ranges for large capital ships (the KGVs had a range of 6000 NM, compared to a range of 15,000 NM for an American South Dakota class) as the RN expected to be able to use the British Empire's wide range of coaling stations. Battleships could also be wet forwards, as they were designed for zero-elevation fire forwards. This would be very useful in a close-range engagement in the North Sea, but would see little use during the war. Comparing the KGVs to their counterparts from other navies gives a good overview of their relative capabilities:

As can be seen, the nearest equivalent to a KGV in tonnage was the American North Carolina class. These had a heavier armament than the British ships, but much less armour and were slightly slower. However, most other navies, and the USN itself, were building considerably bigger ships, with corresponding improvements in capability. The last British battleship to be built was HMS Vanguard, basically an enlarged KGV with 8 15in guns. She was comparable to the ultimate USN battleship, the Iowa class - the Iowas had better armament, but worse armour. British battleships did have several major flaws. Their torpedo protection systems had several major issues, which contributed to the loss of Prince of Wales. The 5.25 in DP gun on the KGVs and Vanguard were rather poor, with their shells being too light for effective anti-destroyer work while being too heavy for a rapid rate of fire for AA defense.

Cruisers reflect much better on the Royal Navy. The various sub-classes of the Town group, and their slightly smaller, cheaper siblings in the Crown Colony class were some of the best cruisers of the war. They were fast, versatile, well-armed and protected, exemplifying the cruiser as a well-rounded warship (in addition, to channel /r/warshipporn for a bit, they were beautiful ships). The smaller cruisers also saw great use during the war, although the Dido class suffered from the poor performance of the 5.25in gun mentioned above.

Royal Navy destroyers tended to be smaller than those from other nations. Even the largest class, the 1850 ton 'Tribals', displaced 200 tons less than the typical American destroyer, the Fletcher class, or the Japanese Kagero class. However, this didn't significantly affect their performance. British destroyer designs used an excellent hull form, giving good stability and high-speed efficiency (though the hull form could have been improved to give better endurance at cruising speed). The RN and Kriegsmarine were the only navies not to lose destroyers to weather during the war (though given how often Kriegsmarine destroyers sortied, the latter isn't exactly surprising or significant). Their armaments varied strongly - some, like the 4in used on the Hunt class escort destroyers, and the 4.7in Mk IX used on older ships, were poor for most uses. Others, such as the 4.7in Mk XI on the more modern destroyers and the 4.5in Mk I used on those entering service towards the end of the war, were excellent guns, comparable to the superlative USN 5in 38 gun. American destroyers were easily the best of the war, but British destroyers were a close second. However, the RN had a clear superiority in ASW escorts. The 'Flower' class, although designed for coastal work, saw sterling service throughout the Battle of the Atlantic, sinking 33 U-boats between 1940-45. While they suffered from heavy rolling, and were rather slow, they were cheap and easy to build (they were based off a design for a whaler that could be built by almost any yard in Britain). The 'River' class of frigates were far better ocean escorts, but were twice as expensive. They were, however, not twice as effective at keeping U-boats from convoys - the much more common 'Flowers' were better at scaring them away, even if they couldn't kill them. The 'Loch' and 'Castle' classes, armed with the Squid ahead-throwing ASW weapon, were the most effective submarine hunters of the war, even if they entered it too late to have a decisive effect.

British shipbuilding tended to the conservative side. This meant that new productive techniques often took time to reach the shipyards, with the 'Lochs' being the first class to be built using prefabrication methods against much opposition from shipbuilders. Director of Naval Construction Stanley Goodall wrote that

[Shipbuilders] don't want bridge builders in the job. I got depressed; it looks as though the builders steady opposition to pre-fab is gaining ground.

However, despite this opposition, British Empire yards, mainly in the UK and Canada, would outproduce all others, barring the US. There were also problems with other areas of shipbuilding. The marine engineering industry was very poor, with British machinery being heavy, bulky, and sometimes unreliable. Leaky piping in the engines of the KGVs were a major case in the reduction of their range to 6000 NM. British boilers required more maintenance and cleaning than those of other nations, primarily because feed-water additives were not allowed. However, despite the profusion of minor problems, no RN ship would suffer a major breakdown during the war.

What are you interested in amount of ships, technology, tactics, strategy, poltical influnce, importance in the war, command?

The best single source on the Royal Navy in World War II in "Engage the Enemy More Closely" by Correlli Barnett. This book looks at the weaknesses of the Royal Navy, such as the fact its naval aviation was decidedly third rate, compared to the Japanese and US Navy. Its anti-aircraft fire control system was inferior to what the US and German Navies used. Barnett really goes off on the outdated and inefficient British shipyard industry, despite the fact the British out built everybody, except the United States, when it came to producing warships during World War II
The real strength of the Royal Navy was it superior training and excellent fighting spirit. Individual British warships may not have been as good as the German ships they encountered, but the British had more ships and were willing to mix it up with Axis fleets in situations that would have caused Axis naval commanders to turn tail and run away.