NOTE: Please don't flood the comments with 'this is how she should have looked', because I have an important announcement; I don't want to talk about what's been done wrong by your standards and I don't care. I'm really sorry to say that I just don't want to discuss things like that. So, would you kindly keep those to yourself?

Adagio's design is a representation of idealized Earth Pony beauty; strong and stocky, with wide hips. She's a representation of a curvier and more plus sized (although not particularly) standard that Earth Ponies have.
Her colors come from both her regular costume and the finale costume. Her gem center and everything around it is based on the gem in her Cutie Mark. Her fins colors come her normal shirt, her shirt in the finale, and her leggings. Her mane antenna is attached to a base of spikes, which are the color of and based on the crown that she normally has in her hair. Her eyelashes were inspired by the sun and generally her appearance is meant to look like the sun.
Adagio is the Superego of the group, ambitious and driven by goals, while not moralizing, she is definitely the one with the most grandiouse and idealized view of herself. She's the brains of the operation, even if Aria would hate to admit it. She's the brains because Aria is much less charismatic and has less control over her emotions, and she has the plans. When they were Water Sprites, she was the one to decide to rise up, and she's continued the trend of leadership for them from the moment they became Sirens to the defeat in Rainbow Rocks. She's very easily frustrated with her cohorts, who are much less oriented towards her dream of ruling, Aria wanting to just rule something, and Sonata only wanting to eat. She has a tendency to snap at them, but out of the three, she is the showmare, the one who knows what she's doing and has the most charisma. Aria is too abrasive to be seductive most of the time, and Sonata is too open to her intentions, and is too dumb to properly do so. So she feels much like she's pulling their dead weight most of the time.
Adagio is my favorite MLP Villain. I will not change my opinion unless somepony or human else comes along and takes it. While she may not be the most evil or most detestable villain, she's my favorite as a character. Her leadership is genuinely entertaining, what with her constant annoyance with her peers, she gets amazing lines, and her voice is gorgeous. She's perfect at sounding like the seductive temptress ideal that Sirens are supposed to live up to, and damned if she doesn't pull it off well. Her songs are amazing and her anger can be both amusing and frightening. I like seeing the brains of the operation much more than I do the... whatever Sonata counts as.