It's amazing to me how many times I've spoken to ambitious DM's excited for their games who can't tell me what sort of campaign they're running. They'll tell me what rules they're running, or they'll try to sum up their plot, but they don't seem to have any thought towards the style of campaign. Rather than just complain I figured since it's campaign month I'll go over some of the basic styles of campaign.
Sandbox: The sandbox campaign, one of my personal favourites, is a GM creating a large world, sprinkling plots and characters and adventures across it, and then releasing the players with a vague goal. If you explain the overarching story of your campaign and at any point you don't know what steps the players will follow to accomplish that goal, you have a sandbox. In a sandbox game the focus is on player freedom, exciting locales and plenty of options for success. The benefit of this type of game is that the players are free to do what they like. The bad thing is the players are free to do what they like. It's important to either be able to improvise well, or to thoroughly plan out all the things the players are likely to encounter or investigate so you don't wind up with an empty box.
Linear campaign: This campaign follows a specific series of adventurers in the hopes of accomplishing a goal. Retrieve the 3 pendants, then get the 7 medallions, then slay the evil wizard. These campaigns sometimes get a bad reputation because they limit player freedom somewhat, but I feel that a nice tight well made campaign can make up for a lot. Because you know exactly what adventures have to get done it's easy to create several very fine, very precisely made encounters. Likewise because you know where the players need to go it's far easier to make fewer but more detailed locations and characters. These games are great for playing strongly on a single theme or idea or fully exploring one or two conepts.
The Oneshot: This campaign is really just a single 1 session long adventure. A one shot is perhaps the most efficient form of campaign from a pure work to play ratio. You need only to make one adventure one or two locales and then you're done, with the added bonus of being able to re-use 100% of it with a new group of players. With that being said, the drawbacks likewise are fairly self explanatory. It's short, the oneshot doesn't generally build anything, it doesn't form a long lasting story, and it's hard for there to be a real deep character arc.
The XL campaign: What happens when a linear campaign has reached the conclusion but no one is ready to stop? The XL campaign ahoy! Sure the world threatening wizard has been stopped, but it's only gonna be about a week or two before The dragon queen rises from the Abyss! And once she's gone the long dead god NarlothOP'fg will awaken. These campaigns are always beautiful to see. When a GM gets to completely cap off a story and then the players want more, that is a great sign that things went well. These games become a series of linear campaigns stacked ontop of each other, but the spackle and grout is in fact the formation of a sandbox. As the world grows, as powers and threats escalate more and more options and allies become factors. You can't start a game expecting this sort of campaign, but many GM's are happy to see it happen. The advantages of this sort of game is that the setting and lore build and builds as the game goes on. This can mean a lot of book keeping, but it can also be deeply rewarding to see.