The Phenomenology philosophy is a school of philosophy that originated in the 20th century. With Edmund Husserl Phenomenology was born but it was the phenomenology Heidegger innovated that reoriented the course of European philosophy. 
The Phenomenology of Husserl and Heidegger (whose respective philosophies are called Transcendental Phenomenology and Existential Phenomenology) does not seek after external objective truth—as philosophy and science generally do—phenomenology strives for subjective truth—in Husserl’s case it is an attempt to make a subjective science of consciousness. Husserl developed a method for attaining this truth which involves reducing away the noise so as to isolate the essence of a phenomenon. Heidegger parted from Husserl saying that phenomenology could not be a science with knowledge because the meaning of a phenomenon is context-dependent. Heidegger moves from Husserl’s epistemological project to an ontological program. 
There is a growing scholarship looking at the connection between phenomenology and eastern philosophy. It is even said that Heidgger was influenced in his conception of Dasein as being-in-the-world through a German commentator on Taoist philosopher Chuang-tzu.
 
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📚 Sources:
Heidegger, M., 2010. Being and time. Suny Press.
Husserl, E., 1999. The essential Husserl: Basic writings in transcendental phenomenology. Indiana University Press.
Smith, David Woodruff, 2018. "Phenomenology", The Stanford Encyclopedia of Philosophy (Summer 2018 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/sum2018/entries/phenomenology/.
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⌛ Timestamps:
0:00 Intro
0:42 History of Phenomenology
1:46 What is Phenomenology
5:59 Husserl’s Phenomenological Method
7:52 Heidegger’s Ontological Twist
9:17 Phenomenology and Eastern Philosophy