Cydir was removed from Dovecot in v2.3.14.
This mailbox format is very similar to Cyrus IMAP's internal mail store:
Messages are stored in "<uid>." named files
Cyrus's cyrus.index is equivalent to Dovecot's dovecot.index. Dovecot however also requires dovecot.index.log for its indexing to work.
Cyrus's cyrus.cache is equivalent to Dovecot's dovecot.index.cache.
Cydir is a very simple format internally. It relies on Dovecot's index files completely for its functionality. If the index files are lost, all the message flags are lost. Currently the code can't even rebuild index files if they're lost.
Cydir is mostly meant to be used for benchmarking and stress testing index handling code. Its code is small and simple, so it can also act as an example for writing new mail storage backends.