A wise man once said:
"The road to hell is paved with adverbs." --Stephen King
The adverb modifies your verb. It's what ensures that you run "quickly", you smile "happily", and you roll down the stairs "jokingly".
Why is the adverb a writing mistake? Well, technically it's not. It's actually an important part of speech in many cases, but it may need to be removed from creative writing as much as possible.
Why is this? Let's take a look at a simple sentence with an adverb in it:
Jane ran quickly through the park.
Nothing wrong with that sentence, right? It implies speed, haste, and moving at high speeds. But why not try:
Jane (sprinted, dashed, raced, tore, hustled) through the park.
Shorter, less wordy, and actually brings to mind an action.
Verbs are strong words. They convey action or a state of being, but in this case we're talking action. When you think of the words "sprint", "dash", or "race", you immediately picture a sprinter like Usain Bolt or Maurice Greene tearing down a track.
On the other hand, when you hear the word "run", you think of a much slower pace, probably closer to jogging than sprinting. Adding "quickly" on the end doesn't really do much for your mental image, just adds another word to the mix.
And that, dear friends and fellow readers/writers, is why the adverb is a weak word nine times out of ten.
I'm sure that we can all come up with sentences where words ending in "-ly" help to make the writing stronger, and if so, good for you! (Imagine me patting you on the head.) But, for the most part, using adverbs just makes the writing weaker and wordier.
Adverbs are like the "passive voice" of writing--using "Bob was listening" instead of "Bob listened". There's something wishy-washy about passive voice (something I'm still learning the hard way), so it's time to trim it from your writing.
Is it really so hard to eliminate all or most "-ly" adverbs from your writing? Truth be told, it's not as difficult as you might think.
"Bob raced quickly" is actually just "Bob raced" (using quickly is unnecessary thanks to the word "raced")
"Wilhelmina Hamburger smiled happily" is actually just "W.H. smiled"
Let's get a bit more complicated:
"He smiled sadly" could be changed to "His was a sad smile"
"He struck the ground forcefully" could be "He pounded his fist into the hard earth"
And so on, and so forth…
In your writing, cut the fluff as much as you can. Adverbs are fluff in most situation, so use your Wolverine Claws of Editing Might to slice them out!