Angels
Angels are, in the most basic sense, divine messengers. Most famously, this refers to Angels of Heaven, as described below. However, it should be noted that while the term "angel" is often associated with the heavenly host, it is technically also a term for any spiritual messenger, and many specific deities have their own "angels" as well. So, while many angels do exist in the hierarchy of heaven, not all do; as Hamlet famously said, "There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy." This holds true, too, for angels.
Angels of Heaven
"Angels of Heaven" in the commonly understood sense are servants, soldiers, messengers, and otherwise keepers of the divine order. They exist to protect creation, shepherd souls, and preserve and maintain Heaven. They act in direct service of The Divine, sometimes described as The Creator or God, but the reality is nothing so simple. In reality, "Heaven" is more of a concept than a singular place--many pantheons have "paradise dimensions" where souls may be sheltered and angels may serve; some of these may have a very Abrahamic, "God the Father" figure, while others follow the pattern of, say, Tír na nÓg. In the end, each pantheon has its own way of doing things, so angels are largely defined by whom they serve.
Fallen Angels
Fallen Angels, sometimes called Demons, are former spiritual messengers, usually servants of a specific deity or of Heaven, who have been cast out that role and, usually, cut off from the divine presence that empowered them.