France had a fairly large, modern navy going into World War II. They had significant overseas colonial interests, and so strong navy was vital to their strategic interests. They were a signatory to the interwar naval treaties (Washington Naval Treaty, London Naval Treaty, Second London Naval Treaty) and were accorded the fourth largest navy in the world (tied for that position with Italy).

On the eve of war, the French Navy Order of Battle was:

2 Dunkerque class fast battleships - these were new (launched in the mid-1930s), interwar production battleships armed with 15" guns. They are sometimes called battlecruisers as they fulfilled a similar role to the British HMS Hood - they could outrun fleet battleships of the day and were intended to deal with commerce raiders (such as the German "pocket battleships").

1 Richelieu class battleship - this was another modern battleship, launched in 1939. Similar in concept to the Dunkerques, it was larger and had heavier armor. It was launched just before the war (Jan 1939) and commissioned in 1940.

1 Bearn class aircraft carrier - this was an early, interwar aircraft carrier, built on an incomplete WWI era battleship hull (this was very common post Washington Naval treaty - the US, Britain, and Japan built similar aircraft carriers). It carried 35-40 aircraft.

5 WWI era dreadnaught battleships - these were WWI era ships still in service. They were all dreadnaughts (Courbet class) or super-dreadnaughts (Bretagne class) that were modernized during the interwar period. This would be similar to Queen Elizabeth class super-dreadnaughts that the British were using (updated WWI ships).

1 seaplane carrier/tender - Commandant Teste. This was another interwar design. The French didn't have another incomplete battleship hull to build another carrier, so this was built from the ground up to support seaplanes (keep in mind that the concept of aircraft carriers as we know it wasn't fully fleshed out at the time).

7 Heavy cruisers - Duquesne, Suffren, and Algerie classes. These were all interwar ships built in compliance with the Washington and London Naval treaties. Displacement of more than 10,000 tonnes and armed with 8" guns

11 Light cruisers - a variety of classes, these were all interwar production and armed with 6" guns. The French sometimes used light cruisers as leaders of flotillas of heavy destroyers, and so these ships were built for speed

32 Heavy destroyers - variety of classes, notably La Fantasque class, which were the fastest surface combatants afloat at the time (they could do 45 knots - for reference this is faster than many warships afloat today). These were consider by the French to be "contre-torpilleur" (anti torpedo boat) - they were designed to be extremely fast to catch small ships that would threaten the main fleet, and very heavily armed. These were all interwar production.

38 Destroyers - variety of classes. These were smaller, conventional destroyers. Interwar production.

80 submarines - variety of classes. Interwar production (including Surcouf.

65 sloops/avisos - these small ships are called sloops in English, though they were "avisos" ("advice boats" or "dispatch boats"). They were for carrying messages and key personnel to the French colonies. Lightly armed and less than 1000 tonnes, they also were used for coastal defense, anti-submarine roles, etc. These would be primarily interwar production as well (Bougainville-class as an example).

The French Navy was a cause for concern in the early war. Before the Fall of France, they operated alongside the Royal Navy against the Germans. When France fell, the disposition of the Navy was a serious concern, as if it fell into German hands it would significantly augment the German Navy overnight (the French Navy was larger at the time). There were also political concerns as some highly placed French naval officers were considered to be sympathetic to fascism. Some French ships escaped to Britain and Egypt, and the British military boarded all French ships in British ports, resulting in some fatalities (Surcouf was one such boat boarded with fatalities). A significant portion of the French fleet, based in Toulon, was scuttled (deliberately sunk by their own crews) in order to keep them out of German hands. Another group of French ships was at Mers-el-Kebir. The British sailed a fleet there and demanded the French surrender or be fired upon. Negotiations broke down and the British attacked, destroying one battleship and leaving most of the rest of the fleet in no condition to to fight.

Edit: Correction - Dunkerque class had 13" guns as the main armament.