Wandering monsters are monsters that can show up in any zone that uses adventure.php, except The Dire Warren and The Shore, Inc. Travel Agency. They can only show up under special circumstances, such as on Feast of Boris, El Dia de Los Muertos Borrachos, or Talk Like a Pirate Day, or Rock monsters during the Unexplained Tremors event, the nemesis assassins, monsters that have been hit with the badly romantic arrow or were winked at, or the monsters during the class revamps.
Wandering monsters all have a fixed window where they can appear, usually with a range of 10 turns. For instance, FoB and El Dia monsters have ranges of 25-35, while monsters hit with the badly romantic arrow have ranges of 15-25. More specifically, the game sets a turncount number equal to your current turncount plus a number in that range, and once your turncount exceeds that set number and you adventure in an adventure.php zone you will get the monster. This means that it is possible to guarantee a wandering monster on a specific turn; you simply have to adventure in non-adventure.php locations (such as the Tavern Cellar).
Wandering monsters are capable of overriding normal combats and noncombats, superlikelies, clover adventures, and others (see Encounter for a detailed explanation), but NOT semirares. This has practical consequences to the adventurer; if you are using dance cards in the Haunted Ballroom, make sure not to use any if a wandering monster has a chance to appear. If you're using a Mechanical Songbird in Dreadsylvania, you'll get the better item drop bonus even though the monster may not be native to the area.
It's not currently known how the game decides which wandering monster you will receive if you can receive more than one. That will be spaded next Feast of Boris, hopefully.
Free runaways (such as those granted from the Stomping Boots) or free combats (such as Bricko Monsters, Hipster Monsters or Infernal Seals) will not change the number of turns needed to encounter a wandering monster.