Blues is meant to make you feel down, Rock is meant to make you want to party, Metal is an aggressive genre, and Pop... Well, I suppose Pop is supposed to make you happy. But is there any specific feeling you're supposed to get from Jazz?
Jazz is often described as an abstract, intellectual genre, so one possible answer would be contemplative. I would say "mellow" is another mood often associated with jazz. Of course, the necessary disclaimer is that jazz is a huge genre with many different variations, and the moods of individual jazz songs run the entire gamut of emotions. But the same is true of all the other genres you mentioned.
It's worth noting that jazz, like classical and perhaps pop, is arguably a less emotional genre. Rock, metal, folk and the blues are all fairly raw, primal and emotional. Jazz, classical and pop are more emotionally distanced, jazz and classical because of their sophistication and complexity, and pop because it's so highly processed and synthesized.
There are tons of jazz subgenres: bebop, free jazz, big band, modal jazz, gypsy jazz, Dixieland, cool jazz...and this isn't close to being a complete list. Given the large variety between subgenres--rather like classical music, in fact--I don't think there is any specific feeling I'm supposed to get from jazz.