Unlike other cultures which employed materials such as papyrus, bamboo, wood, parchment, etc. in their writing culture; Japan, from the offset, used paper.
The introduction of paper to Japan was more or less contemporaneous with the formation of a unified Japanese state. While examples of writing on wooden tablets survive, there was no truly standard method for writing prior to this. This is something to keep in mind when considering the variety of papers used in Japan.
First of all, let us try to define paper. Paper originated in China. The eminent historian of Chinese science Pan Jixing defines paper as follows:
Paper is a medium used for such activities as writing, printing, and wrapping. The process through which it is made can be summarized as follows: first, plant fibers are extracted through a combination of human-made tools and chemical reactions; then, they are soaked in water to obtain a pulp; the water is then drained using a sieve in order to leave the cellulose on the sieve and form a wet paper membrane. After drying, the cellulose bonds together via hydrogen bonding to yield a thin sheet. 
The two operative words here are “plant fibers” and “hydrogen bonding”. Paper can be defined as a sheet of cellulose cells connected together via hydrogen bonding. Plants fibers are not used as they are but first brought to their purest form. The fibers are first separated in water and then allowed to bond together via hydrogen bonding. These two processes are indispensable for making paper. The fibers are not joined together with adhesives but caused to bond together through a chemical reaction.
This is why it is wrong to call materials such as papyrus, which was used for writing in some cultures, “paper”. Papyrus, which was used in ancient Egypt in such books as the Book of the Dead is made from the stalks of the plant. But the stalks are connected via fermentation not hydrogen bonding.
 (Japanese edition: 潘吉星著 佐藤武敏訳 『中国造紙技術史稿』 平凡社 1980／ (Chinese edition: 潘吉星 『中国造纸技术史稿』 文物出版社 1979
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