Four very popular street foods in Korea are 호떡, 핫도그, 꼬치, and 와플. If you spend some time in Korea, it's likely you'll get to try all of these foods.
호떡 is basically like a pancake, but with honey and nuts inside. Eating it reminds me of a churro actually. They're really delicious, but hot, so be careful when eating them.
핫도그 ("hot dog") is basically a double-fried corn dog, so they're thicker than a regular corn dog. The batter is also a bit sweet, and this might surprise you the first time you take a bite. They're also served sprinkled with sugar and ketchup (weird, huh?). The sweet flavor matches pretty well with the sweet crunchy batter, and the ketchup matches with the hot dog inside. Sometimes you can find fancy 핫도그 which have been deep fried together with fries, and these are my favorite kind.
꼬치 is skewered meat on a stick. Basically it's chicken, grilled on a stick, and brushed with sauce while it cooks. Sometimes you'll get other things like green onions on the stick too (these go well with it). The interesting thing about 꼬치 is that their sauces come in many varieties of spicy flavors, ranging from mild ("Did you even put any spiciness on this thing?") to "bomb flavor" ("Why.... Why would you do this to me?") in spicy level. The only time I've legitimately cried from eating any spicy Korean food was when I tried to eat a "bomb flavor" 꼬치 without anything to cool down my mouth.
와플 ("waffle") is a... well, it's a large round waffle. But it's served folded in half, and different ingredients are put inside like a crepe. You can find ones filled with honey and butter (the simplest kind), different jams, chocolate, and even ice cream. There's really nothing like biting into a hot, crispy 와플 while walking down the street on a cold winter morning.
Check out all of these four snacks in the video here.