So a sweater, sweatshirt, and blouse are all just shirts. In America, a sweatshirt and sweater are the same thing, they're long sleeve shirts meant to keep you warm in colder months. A blouse is female clothing with longer sleeves, but isn't entirely made to keep you warm. More of a long sleeve you could wear in spring.
A jumper is a sweater in Britain, but here in America, a jumper is a sleeveless dress worn over a blouse, they were made for each other almost.
As well, in America, pullovers are just sweaters. A sweater with a front opening is just something we'd call a cardigan, if it has a zipper and/or hood, we call it a jacket. Further, depending on the length of the jacket's waist, we call it a coat. But coat and jacket are used interchangeably, no need to be so specific with lengths.
Hoodies are a combo of jackets and sweaters. It's a sweater with a hood.
In short: (Sweater/pullover/British jumper, cardigan, jacket/coat/hoodie) All meant for colder weather, with slight differences, mostly being the presence or absence of hoods and zippers.
(Blouse, American jumper) Fashionable, could be worn year-round, but is mostly worn during spring.