I know IQ stands for intelligence quotient but I have never actually understood what quotient means in this context or where the term came from. I also heard that modern IQ scores are not actually calculated the same way the original quotient was and that the name is kind of outdated. Can someone explain what the term originally meant, how the calculation has changed, and whether the IQ label still makes sense for what tests are actually measuring today?
The history here is actually pretty interesting 
IQ originally stood for intelligence quotient in the literal mathematical sense. The original formula divided mental age by chronological age and multiplied by 100. So a 10 year old performing at the level of a 12 year old would get a score of 120. That formula was developed by William Stern in the early 1900s and popularized by Lewis Terman. The problem is that mental age stops being a meaningful concept in adulthood since cognitive development plateaus, so the original quotient calculation breaks down completely for grown ups. Modern tests abandoned the ratio formula entirely and replaced it with what is called a deviation IQ, which compares your performance to a normed population sample of your own age group and sets the mean at 100 with a standard deviation of 15. So technically the Q in IQ no longer refers to an actual quotient, it is just a deviation score. The name stuck because it was already culturally embedded by the time the methodology changed 
Basically IQ is a historical brand name that outlived its original meaning 
The underlying measurement got more sophisticated but the label never updated. It is kind of like how we still say we are dialing a phone number even though nobody has used a dial in decades.