When spellchecking is turned on, words suspected of being misspelled are underlined in red. Hover over a word to add it to the spelling dictionary.

Interactively for a Single Notebook...

To prevent a word from being marked as misspelled in a notebook, add it to the notebooks spelling dictionary by hovering over the word and choosing Ignore All:

Interactively for All Notebooks...

To prevent a word from being marked as misspelled in all notebooks, add it to the global spelling dictionary by hovering over the word and choosing Add to Dictionary:

Programmatically for a Single Notebook...

Append words to the current notebooks spelling dictionary:

Out[1]=

Programmatically for All Notebooks...

Append words to the global spelling dictionary:

Out[2]=

Notes

Choosing Ignore from an underlined words hover menu suppresses spellchecking for a single instance of a word, but does not add the word to a spelling dictionary:
Misspelled words are highlighted with the setting ShowAutoSpellCheckTrue. Turn off highlighting with ShowAutoSpellCheckFalse.
Words added to spelling dictionaries are stored in the SpellingDictionaries option.