Some of the most common and easiest ways to run an adventure include:
1.) "Dungeon crawls", wherein players explore a hostile environment room-by-room seeking treasure while avoiding or overcoming traps and monsters. The fun in these games comes from a mix of the thrill of discovery and the challenge of staying alive. The simplest possible dungeon crawl is to draw a maze, some monsters in the maze, and some treasure, and then let the players enter the maze while you describe the current room they're in and let them decide where to move and what to do in each room, until they find the treasure and take it out of the maze or die trying. Many DMs got their start as twelve-year-olds running simple dungeon crawls back in the 1970s and 1980s, and it's still a fun way to start DMing!
2.) "Linear plots": stories with a mostly-fixed plot, but with open character roles and opportunities for customization. These are fairly easy to run as long as players cooperate, a bit like a movie where the players can choose who the heroes are and specifically how they overcome the villains of the story: imagine the Twelve Labors of Heracles, but you don't have to be Heracles the strong man. Instead you can be Belfry, an elf cursed with wanderlust who fights monsters by throwing playing cards with the force of shuriken and sneaking through the shadows. You'll still do the Twelve Labors and come into conflict with the Evil king, but the details of how will be different as Belfry than they were for Heracles. Unlike a dungeon crawl, this type of game is often less about challenge and winning/losing than about seeing HOW this particular hero or group of heroes will succeed, and what unique imprint they will make on the gameworld in the process.
3.) "Mysteries" are a bit like dungeon crawls in that the players move around freely seeking something, but unlike a dungeon crawl players usually don't move from room to room seeking treasure, so much as from "scene" to "scene" looking for clues. They require a bit more skill than dungeon crawls because you need to know where to start and stop scenes and how much to skip over: if players say "we go find the police chief and ask him when and where John Thorn was last seen alive," you have to decide whether to start the next scene on the road to the police station (possibly noticing some interesting clue), in the lobby of the police station looking at a police secretary, or in the middle of conversation with the police chief who's now glaring ferociously over his bushy mustache at the player characters after having just barked an answer to their question at them.
In some ways mysteries are still a little like dungeon crawls, but another difference is that in a dungeon you can usually count on players finding everything eventually, including every non-hidden door to every room in the dungeon, but in a mystery there will be many things players never realize so it's very easy to make the mystery too hard unless you build in multiple paths (clues) to every "place" (conclusion, or scene) that the players need to reach.
4.) "Hexcrawls" are like dungeon crawls on a much bigger scale. Instead of exploring a dungeon looking for monsters and treasure, you're exploring a wilderness looking for towns, dungeons, and other interesting places or events (some of which might be linear plots or dungeoncrawls or mysteries). In a dungeon you're usually careful to always describe the exits from each room so that players always know where to go next if they're not interested in the current room ("out the green door" or "through the tunnel to the south"); in a hexcrawl you describe the contents of the local "hex" (i.e. local terrain within a fixed distance such as 5 miles) and then players choose a direction to go (such as "towards the river" or "northwest").
5.) "Social" adventures focus more on personalities and interactions between them than on locations and physical problems. A social adventure has some resemblance to a mystery in how you might navigate between scenes--DM judgment is required to know how much to skip--but the goals are different. Instead of looking for clues, players might seek out conversation which elicits emotional reactions in themselves or other characters (such as confronting an adulterous parent to demand accountability, or pleading with a friend for forgiveness at your inability to get them out of jail).
As before, social adventures need not be wholly separate from other kinds of adventures--perhaps one section of a hexcrawl might be a "home base" of sorts where social situations happen frequently, and even during a dungeon crawl there might be a hidden dwarf civilization on one level where social gameplay is the primary game mode.
Other gameplay modes are possible, such as running a spelljamming spaceship where hexcrawls through interplanetary space lead to dungeon crawls and linear plots whilst managing social issues among the PCs and NPCs who crew the ship. But these five basic types should get you started: dungeon crawls, linear plots, mysteries, hexcrawls, and social adventures.
