You can have up to three dispatch fleets at a time. One is free, which you should definitely use. One costs 300 blue gems to unlock it for a week. One costs 500 red gems to unlock it for a week. All three also have company level requirements.

When a dispatch fleet is unlocked, you can fill it with as many ships as are allowed in your main fleet. Any ships can be used, but the critical thing is to be able to load as many mates onto your dispatch fleet as possible. Ship bonuses to various stats can also make additional dispatch options available.

As of this writing, there are 112 dispatch options available in the game. It costs some ducats to send a fleet away on dispatch, and then after a fixed number of hours, you get the rewards. Each dispatch option is associated with either adventuring, trade, or combat. Trade dispatches will generally reward you with massively more ducats than it cost to dispatch the fleet, but only give you ducats. The other two will reward fewer ducats than it cost to dispatch the fleet, but also give other types of rewards. Most importantly, adventuring dispatches will give ship parts, while combat dispatches give mate gear.

All three reward mate experience of the type that corresponds to the dispatch. This allows you to use dispatch to level up mates, and then those higher level mates can have improved stats that unlock higher level dispatches. Alternatively, you can swap mates between your dispatch fleet and main fleet whenever your main fleet is in port and the dispatch fleet is not on dispatch.

Some combat dispatches with a particular character's name will award tradeable contracts for the named mate. For example, Cornaro Assault awards contracts for Catherine Cornaro. This can allow you to promote and train particular mates all the way up to S or SS grade without needing to draw one particular mate a large number of times in the gachas. It is also the source of the large stacks of contracts available cheaply on the auction house.

The particular dispatch options are all listed in game, but in a way that can make it difficult to find what you're looking for. I've sorted then by type below, and then within each type, they are sorted roughly from lowest level requirements to highest. Most require the same stat level for each of the four stats of a given type, but some require slightly higher values of one stat than another. I've listed the latter as using some non-integer stat level, then explain the meaning of values below the main table.

For adventuring and trade, the stat level is the fleet stat as displayed on the fleet screen. For combat, they take the sum across all ships in the fleet, then divide by the number of ships, and use the level that that would give on a single ship.

For the half integer stats above, add 0.5 for nature and scouting, and subtract 0.5 for aesthetics and supply.

The meaning of the non-integer stats values above is: