Spell check underlines a misspelled word as you write, on desktop and on mobile. It starts off, so a note full of names, tags and code stays free of red lines until you ask for them.
Turning it on
Open Settings, go to Notes and switch on Spell check, on desktop or on mobile. It is one setting: flip it on either device and sync carries it to the others.
On desktop
Misspelled words get a red underline. Right click one to see suggestions, pick one to replace the word, or choose Add to dictionary so that word is never underlined again. That is the place for names, slang and technical terms the dictionary does not know. On Mac and Windows the word also goes into the system's own dictionary, so other apps stop flagging it too.
To take a word back out, right click it and choose Remove from dictionary. On Windows and Linux select the word first, since only the Mac selects it for you on a right click.
On Mac the system spell checker is used, so it follows the languages set in macOS and recognizes the language a sentence is written in. On Windows and Linux the dictionary for your system language is downloaded the first time it is needed, so the first underline on a fresh install can take a moment.
On mobile
The keyboard does the checking, so it works the way it does in every other app on your phone: misspelled words are underlined, and tapping one shows corrections. The languages come from your keyboard settings. Autocorrect is a separate keyboard feature and keeps behaving the way it did before, whether spell check is on or off.
Turning it off again
Switch it off in the same place and the underlines go away on every device. Nothing about the text changes either way, spell check only marks words, it never rewrites them.