In Faxiang 法相 doctrine, bodhisattvas are distinguished into suddenly awakened 頓悟菩薩 and gradually awakened 漸悟菩薩. Suddenly awakened bodhisattvas have ripened faculties which allow them to arouse the great intention for enlightenment 大菩提心 from the outset, enter directly into Mahāyāna practices, and achieve Mahāyāna enlightenment. Because they do this without hesitation or wavering, they are also called 'directly proceeding bodhisattvas'  直往菩薩. Gradually enlightened bodhisattvas are bodhisattvas of indeterminate nature 不定性 who start by attaining the arousal, practice, and realization of the śrāvaka 聲聞 or pratyekabuddha 獨覺, and then afterward awaken the intention for the enlightenment of the Great Vehicle 廻心向大.

In the Tiantai 天台宗 doctrinal taxonomy, sudden and gradual constitute the first two of the four modes of teaching 化儀四教, with the sudden teaching 頓教 being the approach that guides, without reserve, giving the whole truth, specifically identified with the Huayan jing 華嚴經. The gradual teaching 漸教 includes the content of the Āgamas 阿含, Vaipulya 方等, and Prajñāpāramitā 般若 sūtras.

In the Chinese Chan tradition, the contrast between the two is articulated most directly the Platform Sutraʼs 六祖壇經 story of the confrontation between the two candidates for Dharma transmission of the fifth patriarch Hongren 弘忍—Shenxiu 神秀 and Huineng 慧能, wherein Huineng attains the authoritative transmission through the composition of a verse describing enlightenment in sudden terms. Shenxiuʼs verse, which emphasizes the need for continual polishing of the mind (represented by a mirror), is emblematic of the gradual position, whereas Huinengʼs verse argues that even the notion of a mirror lacks basis, so what can be there to be polished?

Within Chinese Chan, the issue of sudden and gradual enlightenment is again taken up at length in the Sutra of Perfect Enlightenment 圓覺經, where the topic is discussed from a variety of perspectives through several chapters. This sutra clearly valorizes the sudden approach,1 but still allows for the gradual approach at the end, with the author apparently understanding that even if the sudden approach is ultimately the only valid one, the mental subtlety required to properly grasp it is not something developed in all practitioners. In the subsequent history of Chan in China, the sudden approach becomes a basic standpoint of the Linji school 臨濟宗, with its gong'an-oriented practice 公案 based on an assumption of a basic suddenistic standpoint.

In Korean Seon, the issue of sudden and gradual was taken up repeatedly as a topic of discussion and debate, with the early Korean Seon teachers emphasizing the sudden view based on Chinese influence. In the Goryeo period, the influential Seon teacher Jinul 知訥 (1158–1210) advocated 'sudden enlightenment followed by gradual practice,'  頓悟漸修 saying that the initial awakening experience had to be sudden, but that since peopleʼs minds are embedded deeply with defilement, it still takes effort toward purification to attain perfect Buddhahood. While this position was accepted by many, it was also criticized from time to time, and in the 20th century, the eminent Haeinsa master Seongcheol 性徹 (1912–1993) argued against Jinul, saying that enlightenment could only be attained through a thoroughly sudden approach, thus denying Jinulʼs valorization of gradual practice.

References:

Seongcheol. 1987. Echoes From Mt. Kaya: Selections on Korean Buddhism. Seoul:  Changgyŏng-gak.

Buswell, Robert E., Jr. 1987. “The 'Short-cut' Approach of K'an-hua Meditation: The Evolution of a Practical Subitism in Chinese Ch'an Buddhism.”  In Sudden and Gradual Approaches to Enlightenment in Chinese Thought. Honolulu:  University of Hawai`i Press. 321–77.

Faure, Bernard. 1991. The Rhetoric of Immediacy: A Cultural Critique of Chan/Zen Buddhism. Princeton:  Princeton University Press.

Gregory, Peter N, ed. 1987. Sudden and Gradual: Approaches to Enlightenment in Chinese Thought. Honolulu:  University of Hawai`i Press.

McRae, John R. 1986. The Northern School and the Formation of Early Ch'an Buddhism. Honolulu:  University of Hawai`i Press. Studies in East Asian Buddhism3,

McRae, John R., trans. 2000. The Platform Sutra of the Sixth Patriarch. Berkeley, California:  Numata Center for Buddhist Translation and Research.

Muller, A. Charles. 1999. The Sutra of Perfect Enlightenment: Korean Buddhismʼs Guide to Meditation (with the commentary by the Sŏn monk Kihwa). Albany, New York:  SUNY Press.

Park, Sung Bae. 1983. Buddhist Faith and Sudden Enlightenment. Albany:  SUNY Press.

----. 1994. “On the Subitist/Gradualist Debate in Korean Buddhism: Sŏngch'ŏlʼs Theory of Sudden Enlightenment and Sudden Practice.”  In Suh, Dae-sook, ed. Korean Studies: New Pacific Currents. Honolulu:  University of Hawai`i Press. 163–176. Pacific Association for Korean Studies

Yun, Woncheol. 1994. “On the Theory of Sudden Enlightenment and Sudden Practice in Korean Buddhism: Texts and Contexts of the Subitist Gradualist Debates Regarding Sŏnmun Chŏngno.” Stony Brook. State University of New York.