Short answers, pulled from the story.
Edmund Husserl founded phenomenology as a distinct philosophical movement at the beginning of the 20th century. He drew on the work of his teachers Franz Brentano and Carl Stumpf, particularly Brentano's theory of intentionality, to develop a method he described as the "science of experience." The term first appeared in direct connection to Husserl's philosophy in a 1907 article in The Philosophical Review.
Intentionality in phenomenology refers to the idea that consciousness is always directed at or "about" something, whether a physical object, a memory, or a fantasy. The term originated with medieval Scholastic philosophers, was revived by Franz Brentano, and was refined by Husserl into the cornerstone of his theory of consciousness. Husserl specified that intentionality is an intrinsic feature of conscious acts, not a relation between two independent terms.
Heidegger shifted phenomenology's focus from consciousness to existence itself, introducing the term Dasein to name a mode of being that cannot be reduced to consciousness. He replaced Husserl's concept of intentionality with the notion of comportment, and argued that practical, everyday "know-how" is more fundamental than the reflective acts Husserl analyzed. Husserl charged Heidegger with abandoning genuine phenomenological inquiry in favor of abstract anthropology.
Phenomenologists Shaun Gallagher and Dan Zahavi identify four basic steps: epoché, the suspension of commonsense and theoretical assumptions; phenomenological reduction, which traces correlations between experience and the structures of subjectivity enabling it; eidetic variation, the imaginative process of determining what features are essential to a thing; and intersubjective corroboration, the sharing of findings with a research community to distinguish individual idiosyncrasy from universal structures of experience.
Noesis refers to the real, live part of a conscious act, such as judging, perceiving, or loving, while noema refers to the ideal content or meaning that gives the act its sense. Husserl introduced the distinction in his 1913 work Ideas: General Introduction to Pure Phenomenology. The full noema comprises at minimum a noematic sense and a noematic core, and the precise relationship between the noematic object and the actual object of an act has been a long-running controversy in the tradition.
The lifeworld, Lebenswelt in German, is the pre-theoretical background or horizon against which every object appears as distinct and meaningful. Husserl described it as both personal and intersubjective, calling the shared version a "homeworld." It is the ground that science and philosophy both presuppose but rarely examine directly.