What are Arachin ערכין and how does one figure out their value? In a similar vein, what are Nedarim נדרים and how are they figured out? And finally, what is the difference between these two items?
Erechin are described in Leviticus 27:1-8, and the idea behind them is discussed more here. The Torah assigns a simple "erech-value" to people based on their age and gender. Thus if someone pledged to donate "the erech of so-and-so", s/he would look up their age and gender in the erech table and be obligated to give that amount of silver.
A neder means a person vowed to bring a sacrifice. (See Rashi, Leviticus 22:18). So if a person said they hereby made a neder to bring, let's say, a shelamim sacrifice, s/he would be obligated to bring some appropriate animal as a sacrifice.
At face value, it would seem that the mitzvah of arachin is ‘powered’, so to speak, by nedarim. It is a pledge to pay one’s ‘eirech’ value, not unlike the neder one can make to donate one’s ‘damim’ value, i.e. his worth as a slave, to hekdesh. Despite their similarities, however, there is a fundamental difference.
The Chazon Ish, in his introduction to Maseches Arachin, puts forth a yesod which transforms the simple understanding of the mitzvah. Here are his words: ונראה, דענין הערכין אינו חייב מדין נדר את הסך הקצוב אלא ש"נתפש" במצות הערכין, וכענין התפשת נזירות.
The idea of Arachin is not a glorified neder; rather, a person gets ‘taken’ by the mitzvah of arachin, much like a nazir is ‘taken’ by his nezirus. In other words, arachin is the process by which a person is makdish himself to Bedek Habayis, and therefore needs to be redeemed. That money is his eirech value. This is like a nazir, whose neder launches him into a whole new reality called nezirus. He becomes a different person. So too when one is ma’arich himself, he becomes the property of hekdesh, as it were, and thus needs to redeem himself.
A simple nafka mina is bal yachel. A person who violates his neder transgresses the Torah prohibition of לא יחל דברו. A nazir who drinks wine, however, is not. Even though his nezirus is technically ‘powered’ by a neder, his new identity as a nazir places him in a new category, and he would only transgress the issurim directly stated in the parsha of nezirus. The same is true, says the Chazon Ish, by arachin. Because the obligation to pay is not a direct result of the neder, but rather of the pidyon that accompanies his newfound hekdesh status, failure to pay does not violate bal yachel.
So to answer your question, nedarim are a person's value as a slave, whereas arachin are a person being makdish 'himself', as it were, to hekdesh, and then subsequently being podeh himself for a preassigned value.
As to how the values are determined, Rav Hirsch deals with it, and it was written on another post on Mi Yodeya here.
See also Alshich and Shiltei Giborim below for further sources.
אלשיך: וההקדמה היא כי יפליא נדר והאמירה שאחריו הוא בערכך נפשות לה'. והוא לומר הנה דע לך כי ענין הנדר אי אפשר שיהיה לשום בחינה כיוהרא שהוא וותרן וכיוצא. רק על בחינת הפלאה שהוא לפרוש עצמו משטף דברים גופניים ולהתקדש לשמו יתברך. כי על כן אמור ערכי עלי למען תחול קדושה על עצמו מעין כל דבר שהוקדש ונפדה. שלא יבצר מלחול בו קדושה מהכין זה לו הכנה להנהיג עצמו בקדושה. כאומר בלבו איני אשר הייתי עד כה. כי עתה הוקדשתי לשמים ונתתי ערכי לה'.
שלטי הגבורים ע"ז דף ד. בדפי הרי"ף: אינו נ"ל קושיא די"ל דגם האומר ערכי עלי הוי מקדיש שפיר כלי ידוע דהיינו גופו ועצמו שהוא מקדישו אלא שפודהו אח"כ מיד הקדש בדמי ערכו ודמי למקח וממכר...