Body language is an element of character because it's not usually possible to control it, especially when under the influence of strong emotions. Some people can subdue it, or manage not to react to things by eliminating their reactions overall, but in a story where the body language of the characters is written, their very blankness may stand out.
Because body language is uncontrollable, characters can often reveal things through their actions even if they're unlikely to talk about them, or even if they're not entirely aware of them. For instance, someone who's talked into doing something they don't really want to do might exhibit nervous body language, even when they're verbally in agreement.
Body language--especially gestures, posture and movement--can also speak to people other than those who are interacting because they're visible to an observer as well. That makes them particularly valuable to spies, watching the exchange, but also to readers. If you describe the scene and the body language, those nonverbal cues can convey the feel and mood of the conversation or interaction. Posture and movement become an important part of characterization and creating subtext for the reader.
It's especially important to describe a character's body language when two or more characters are interacting. While a detail about a lone character's posture or positioning can help to set the scene and communicate a character's mood, body language comes into play the most clearly when characters are interacting. Not only that, but a description of body language during character interaction adds depth. It can communicate subtle disagreement or discord between the characters, reflect their feelings for one another, or indicate deception or miscommunications.
In mysteries plots, body language can be used to lay clues or foreshadow later revelations, and in fantasy, science fiction and horror there are some wonderful ways to play with body language. Different species might well use it differently and in different ways. For most humans, the visual sense is the dominant one, which is probably part of the reason that visual nonverbal cues tell us the most. We're evolved to look for them, rather than listen for them, or smell for them.
I don't think body language is used as much in writing as it could be. We often describe character's postures and positions in very general terms--sitting at the table, standing in the doorway, leaning against the railing--without including the most important details. While I don't think that every body position and posture should be described in perfect detail, simple things like a mention of crossed arms, bitten lips, or the rubbing of a forehead or ear can speak to the reader.
Subtle cues may need to be repeated, but they can form a quiet web of signals and signs that create an undercurrent. And when a given character says something with a given body movement, which is repeated throughout a text, it can become good characterization. That level of detail isn't always necessary--or desirable if something needs to be clearly understood by the reader--but it can add to the mood or the tension in sections where those elements need strengthening.
Personal space is another aspect of body language that changes even between human cultures. While Americans prefer about three feet of space between themselves and people they don't know well, other cultures might find the need for so much space to be rude. The relationship of the characters and their emotional states will also affect how much personal space they give each other, and how much they insist on having. This can make for interesting reading, as a character who requires more space than others might be considered distant or rude, even when what they say is polite or civil. A character who feels too much space is rude can be talking to someone and all but chasing them across the room as they move closer and the other person puts distance between them.
Gestures, hand signals and facial expression are all culturally influenced and can create depth when you're worldbuilding. Cultures have rules about when it's okay to show a given emotion (anger, sorrow, joy) and who can express what. A man in America expressing sorrow through tears would be considered odd in many cases. This not only affects the characterization of the character, but of the society. In a place where it's perfectly normal for people to show sorrow when they are hurting, you might have character weeping in the street without even drawing the attention of those around them. Or, if one is expected never to show sorrow--and thus force it on those around them--then that character would be drawing a lot of attention. Which someone else might then be able to use as a distraction.
Of course, some people are harder to read than others, and on an individual by individual basis there will be a difference in how the nuances of emotions are expressed even within a given culture. This adds another level of complexity to the dialogue, allowing characters to deceive one another while their deception is perfectly clear to the reader.