** Emacs now supports on-the-fly spell checking by the means of a
minor mode. It is called M-x flyspell-mode. You don't have to remember
any other special commands to use it, and you will hardly notice it
except when you make a spelling error. Flyspell works by highlighting
incorrect words as soon as they are completed or as soon as the cursor
hits a new word.
Flyspell mode works with whichever dictionary you have selected for Ispell in Emacs. In TeX mode, it understands TeX syntax so as not to be confused by TeX commands.
You can correct a misspelled word by editing it into something correct. You can also correct it, or accept it as correct, by clicking on the word with Mouse-2; that gives you a pop-up menu of various alternative replacements and actions.
Flyspell mode also proposes "automatic" corrections. M-TAB replaces the current misspelled word with a possible correction. If several corrections are made possible, M-TAB cycles through them in alphabetical order, or in order of decreasing likelihood if flyspell-sort-corrections is nil.
Flyspell mode also flags an error when a word is repeated, if flyspell-mark-duplications-flag is non-nil.