Within Sea of Thieves, you will receive receive quests from each of the three main Trading Companies, the same that are commonly referred to as Voyages. Not only will you earn gold by successfully completing these Voyages, but you will also gain Reputation for each of the Trading Companies as you do, getting you one step further to becoming a Pirate Legend. Though each Voyage will be different from one to the next with each of the three Trading Companies, the general task will remain basically the same in regards to the Voyages one Trading Company will give you.
When you first begin to undertake Voyages for a Trading Company, you will be able to purchase said Voyages for free. After you complete a number of Voyages and your Reputation increases with a Trading Company, you will eventually be able to buy promotions that will see you being able to take on even harder quests. The more expensive this gets, the better your rewards will be for completing a Voyage. The degree of difficulty for a Voyage can also be determined simply by the terminology used for it.
Even after receiving a Voyage from either the Gold Hoarders, the Order of Souls or the Merchant Alliance, you will still have to vote with your crew on whether to undertake it. If you're doing a Solo Run of the game, all you will need to do is place the Voyage Scroll on the table opposite that of the main map on your Ship, voting whether of not to proceed with said Voyage. If you're playing with others, each member of the crew will actually have to place their Voyage Scroll on the main map table, and then the entire crew must all vote for which Voyage they would like to proceed with (any Voyage not voted on will return to your Inventory).
Depending on which Trading Company you are completing Voyages for, you quests will contain different aspects and will range in difficulty: the Gold Hoarders will task you with finding buried treasure; the Order of Souls will task you defeating skeleton armies and claiming the Bounty Skulls of Skeleton Captains; and the Merchant Alliance will task you finding and delivering trade goods in a timely fashion. If you leave the game at any given time during a Voyage or are booted out due to inactivity, you will abandon any and all Voyages you are presently undertaking and must start over again.