Abstract We report the properties of more than 600 bursts (including cluster-bursts) detected from the repeating fast radio burst (FRB) source FRB 20201124A with the Five-hundred-meter Aperture Spherical radio Telescope during an extremely active episode on UTC 2021 September 25–28, in a series of four papers. The observations were carried out in the band of 1.0–1.5 GHz by using the center beam of the L-band 19-beam receiver. We monitored the source in sixteen 1 hr sessions and one 3 hr session spanning 23 days. All the bursts were detected during the first four days. In this first paper of the series, we perform a detailed morphological study of 624 bursts using the two-dimensional frequency-time “waterfall” plots, with a burst (or cluster-burst) defined as an emission episode during which the adjacent emission peaks have a separation shorter than 400 ms. The duration of a burst is therefore always longer than 1 ms, with the longest up to more than 120 ms. The emission spectra of the sub-bursts are typically narrow within the observing band with a characteristic width of ∼277 MHz. The center frequency distribution has a dominant peak at about 1091.9 MHz and a secondary weak peak around 1327.9 MHz. Most bursts show a frequencydownward-drifting pattern. Based on the drifting patterns, we classify the bursts into five main categories: downward drifting (263) bursts, upward drifting (3) bursts, complex (203), no drifting (35) bursts, and no evidence for drifting (121) bursts. Subtypes are introduced based on the emission frequency range in the band (low, middle, high and wide) as well as the number of components in one burst (1, 2, or multiple). We measured a varying scintillation bandwidth from about 0.5 MHz at 1.0 GHz to 1.4 MHz at 1.5 GHz with a spectral index of 3.0.