A. You mistakenly deleted a chunk of text in an entry
If the entry is still open and you have not yet tapped Done: tap Edit in the Mac menu bar, then tap Undo.
If you have already tapped 'Done' to exit the entry: there is unfortunately no way to recover the deleted text.
B. You mistakenly deleted an entire entry
Open the Trash folder in a Finder window, and look for a file with the .doentry extension. The full filename will be something like this:
Right-click on that file, then open it with TextEdit. In the file that opens, you will see a lot of unintelligible meta-data; the actual text of your entry will appear intact in the middle of the meta-data. Locate the text, copy it, and paste it into a new entry in Day One.
If you have already emptied the Trash: the file is likely not recoverable. However, those who sync to Dropbox might be able to find the entry via Dropbox's method for recovering deleted files