With the default settings, the app will download up to 25 messages at a time, and cache on the device up to 250 messages (pruning as necessary over time).
The relevant setting are:
For how many to sync at a time -- "messages per page" in global app settings under "mail, receiving" -- or per-account, long press -> options and folders -> account options.
For how many to keep on the device -- per-account, long press -> options and folders -> account options