Is there a difference between a חֵטְא (cheit) and an עֲבֵרָה (aveira)? I translate both of those words as "sin." Maybe they signify a different level of magnitude? Or maybe one word is simply applied to one type of sin while the other is applied to another type? Maybe there is no difference at all?
I found a quote from the Malbim in Ayelet HaShachar, I couldn't find the original online (according to wikipedia it was published as part of HaTorah VeHaMitzvah). Ayelet HaShachar is a linguistic guide to similar words.
[ חטא עון פשע ] פעל חטא נאמר על הנטיה מדרך הראוי וכולל בין השוגג בין המזיד כל שלא היה בסיבת כפירה ומינות, ועל כן הוצרך לאמר כי תחטא בשגגה ( ויקרא קצב ויקרא שמח ), אבל אם באו השמות בדיוק וכ"ש כשנרדפים ידח, יציין בשם חטא את השוגג, ובשם עון את המעוה מצד השכל שהוא המזיד. ובשם פשע יציין את המורד בשאט נפש ( ויקרא קצב אחרי כא אחרי נה ), ולפי זה המעוה הוא הפך השוגג. והפשע הוא הפך האונס. והחוטא הוא הפך הספק ( אחרי נה ):
In short (assuming I understood it correctly):
"חטא" - means deviating from the right path, whether on purpose or accident. As long as it was not done as an act of denying or heresy.
"עון" - someone intellectually decides to sin. i.e. on purpose
"פשע" - someone who rebels in disgust (see Yechezkel 36:5)
Based on this, I would posit that "עֲבֵרָה" just means a sin, in general. While "חטא", "עון", and "פשע" are particular styles of sin.
I addresses this question in a relatively recent essay from earlier this year (see here, also published here). Basically, the Talmud (Yoma 36b) explains that chet refers to an inadvertent sin (the state of mind known as shogeg), avon refers to wanton/intentional sins (meizid), and pesha refers to sins of rebellion. Other sources say that chet means "lack" and may refer to the lack of fulfilling a positive commandment. An earlier answer essentially summarized these points. All of these words differ from the term "aveira" in that they all appear in Tanakh, while the word aveira and its cognates first appear in rabbinic sources. "Aveira" is related to the root עבר "pass" which refers to somebody "passing" or "crossing" over a certain red line without specifying that he had sinned, per se.
SOURCE: What's in a word?, "Degrees of Sin"
Pesha-intentional sin to make Hashem angry (kivyachol)
Source: HaRav Yosef Mizrahi Shalit"a, and Siddur Kawanat HaLev.