Octavia Silver is from a newer pureblood family. I imagine that the Silver family is old compared to most, but is shunned by many of the older "Sacred Twenty-eight". The family is extremely ambitious. Octavia's father "married up", albeit into a French family, and he expects Octavia to forward the family as well. He arranged a marriage for her when she was still at Hogwarts (Ravenclaw) to a member of the Parkinson family. She refused to comply and was basically disowned.
Now she's back at Hogwarts a healer and professor of healing arts. To help the Order, she is attempting to get into the good graces of many of the pureblood students. She wants to show them that they can forge their own paths.
Because Octavia was raised to be a very proper pureblood lady, she tends to dress a bit more formally than most people. She has a sort of updated pureblood style. It has some retro and some Victorian influences, but it is stremlined a bit. She very rarely wears pants because she is quite short and thinks skirts make her look taller.