First, adverbs and adjectives are two different kinds of descriptive modifiers and should not be confused with one another. Adjectives modify nouns ("a red ball"). An adverb usually modifies a verb ("he ran quickly") and sometimes an adjective ("he was incredibly smart"). As a general rule, adverbs are best used sparingly. For example, if a strong verb is used, it may not require a modifier. Thus, instead of "he ran quickly" you can more effectively write "he sprinted."
Your question is perhaps better posed as, "How can descriptive language be used to develop a story's character?" Descriptive language, after all, can include verbs. For example, "He shuffled into the room" is more descriptive than "He walked into the room."
Descriptive language is pivotal in developing a story's character. It can be used to bring to life three different aspects of a character:
1) their appearance
2) how they speak and act
3) how they see the world
The first two are fairly obvious and straightforward, but the third is more difficult and elusive. Basically, every character - depending on their age, gender, class, occupation, etc. - will see the world through a different lens. Thus, the language used to describe their thoughts and perceptions should reflect that particular lens.
For example, say you are describing a character walking into a room for the first time. If your character is an interior designer, he may notice the wallpaper and the color scheme. If he is a photographer he may notice how the room is lit. If he is a writer he may look to see what books are on the shelf. These are all subtle ways of showing rather than telling who your character is.