The spell checker function in the new version of Moodle 2.5 has actually been altered to rely on the in-browser spell checking functionality.
The previous 'button' method of the spell checker is now known as the 'legacy spell checker', however this version of the spell checker will only appear in Internet Explorer 9 or earlier.
Using any browser other than IE 9 or earlier, users should notice red underlines appearing beneath misspelled words in the text editor, and should then be able to Ctrl-right click on the word to utilize the browser spell check.
If a user just right clicks on the word the spell check option will not appear, but the window will instruct them to ctrl-right click for more options.
Please see image below,