How can I add a spellchecker to a richtextbox in my application?
You can purchase a spell checker control, integrate the Microsoft Office Spell Checker, write your own (which isn't too hard, actually, once you get the Soundex function figured out), or get a good free one. Here's a (relatively) good free one.
http://www.codeproject.com/KB/recipes/spellchecker_mg.aspx
For commercial products, I'd say to Google "Spell check WinForms"
If you're interested in rolling your own, I wrote one for Asp.Net back when I was in my beginner phase, and even then it only took me about a week to research and then a day or so to code. It's a fun pet project. I'd start by looking at soundex functions and comparing soundex values for similarity.
You start by comparing all of the words in the TextBox to a known dictionary, and using the soundex functions to come up with similar words.
From there, you can go on to creating a table for "popular replacements. (for example, you can track that the word "teh" was replaced by "the" n times and move the more popular replacements to the top of the list.