With the Buddha teaching for 45 years before he passed into parinibbana, what is known about the methods and practices used to keep his teachings alive until they were written down? Do we know how many people were involved in this effort? How was such an extensive collection of teachings categorized?
Traditionally the original source of the teachings is Ananda. Ananda was the Buddha's cousin and primary attendant during a lot of his life up to the Buddha's death. Being so close to the Buddha, he heard at first hand the Buddha's teaching and was able to commit them to memory. Shortly after the Buddha's death the First Council was convened and Ananda was prevailed upon to recite all the Buddha's teachings he could recall. Fortunately for us he could recall a lot.
This is why many of the Pali Canon sutras begin with 'Thus have I heard'. It is Ananda speaking and recalling the Buddha's teachings from his own experience.
Example from the Mangala Sutta
Thus have I heard. On one occasion the Exalted One was dwelling at Anathapindika's monastery, in Jeta's Grove, near Savatthi. Now when the night was far spent, a certain deity whose surpassing splendor illuminated the entire Jeta Grove, came to the presence of the Exalted One and, drawing near, respectfully saluted him and stood at one side.
This is a pretty broad question, so this is an attempt at a broad, but not very deep answer. Hopefully someone else will lend a hand with sources and/or corrections.
I believe the oral tradition (of memorizing and repeating long texts) was solidly established in ancient India before the Buddha's time, as a way of becoming authoritative in a field of knowledge. You can still see this in other extant Indian religions.
As for the number of people, several hundred monks were present at the Buddha's death, and from the very beginning of his ministry the Buddha told his best students to go off in different directions and teach. So we can assume there were thousands upon thousands of monks participating in the oral tradition throughout ancient India.
But anyway, the Pali Canon itself was written down in Sri Lanka. I wish I could remember where I read this. Other canonical texts, such as the Chinese texts have a lot in common with the Pali Canon.
The teachings are categorized into three baskets: Training rules (Vinaya Pitaka), Doctrine (Sutta Pitaka), and Metaphysics (Abhidhamma Pitaka), with the Doctrine basket being further categorized by length. You can see this categorization scheme at:
The Sutta Pitaka is the most relevant section to laypersons looking to inform their practice.
The way the Pali Canon was transmitted by oral tradition, is consistent with the way the Vedas was transmitted for thousands of years. The composition is in a poetic form, using mnemonic formulae for memorization and recitation. The recitation would have also served a ritualistic or ceremonial purpose. Even the tradition says that in the First Buddhist Council, Ananda and Upali recited the suttas and the vinaya. Even if they did not recite in Pali, they would have recited in a Prakrit dialect that's very similar to Pali.
Please see "Pali Oral Literature" by L.S. Cousins. I quote below:
Early Buddhist literature is an oral literature. Such a literature is not without its own characteristic features. A widespread use of mnemonic formulae is one of the most typical of these. I would refer to the considerable body of research on the nature of oral epic poetry. In such poetry the formulae are used both as an aid to actual performance and to maintain the continuity and form of the epic tradition.
Both these features are certainly present in the sutta literature. In the first place many suttas are clearly designed for chanting. We should assume that, then as now, their chanting would produce a great deal of religious emotion - the pamojja and piti-somanassa of the texts. The difference of course would be that the language of the suttas would still be directly comprehensible to the hearers. In these circumstances suttas would be chanted by individual monks both for edification and for enjoyment. We may compare the recitations attributed to Ananda and Upali in accounts of the First Council. In practice they would have to be tailored to the needs of the particular situation ~ shortened or lengthened as required. An experienced chanter would be able to string together many different traditional episodes and teachings so as to form a coherent, profound and moving composition.
Scholars regard this language (Pali) as a hybrid showing features of several Prakrit dialects used around the third century BCE, subjected to a partial process of Sanskritization. While the language is not identical to what Buddha himself would have spoken, it belongs to the same broad language family as those he might have used and originates from the same conceptual matrix. This language thus reflects the thought-world that the Buddha inherited from the wider Indian culture into which he was born, so that its words capture the subtle nuances of that thought-world.