There are two concepts in the Halacha of Bittul (nullification). One says that if there is an item that is Unkosher that falls into another item that is Kosher, and it is a mixture, and the Unkosher is one-sixtieth, it is nullified and you can eat the mixture even though it has the Unkosher in it. However, there is another Halacha along the same premise that says if Roiv (a majority) of the food is Kosher, then the Unkosher food is nullified. Wouldn't you know that from the fact that it is more than SIXTY?
There are really two types of Bittul (nullification):
Min BeMino - a mixture of the same types of food. In this case the prohibited substance (the 'issur') is batel (nullified) if it is in the minority (rov). The rabbis enacted a restriction (gezeira) not to eat the mixture until the issur is less than one 60th of the total (shishim).
Min BeSheEino Mino - a mixture of different types of food. Here, the issur is batel if it no longer imparts flavor (noten taam). To determine this, one would have to have an expert chef (kefeila) taste the food to determine if the flavor is there. Since this is not always practical (and we assume we are not experts at 'tasting') we wait until shishim and then assume the flavor is gone, unless it's a very flavorful food (tavlin) when we usually assume you are just stuck.
(There are way too many sources, but see the Taz YD 98:3 for a similar, succinct, and good enough summary.)
Please make sure to CYLOR before relying on these principles yourself, as there are many small details that can make a difference.
Sometimes rov is enough and other times we require shishim. See this chart for details but basically the only case l'halacha we will rely on rov is if no taste is transferred (yavesh byavesh) and taste the same (min bmino).
To quote myself:
The answer, explains the Ra'avad (quoted by the Rashba to Chullin 89b ד"ה אמר), is as follows:
כי אמרינן חד בתרי בטיל מדאורייתא דוקא דקיימא איסורא באפי נפשה והיתירא באפי נפשיה כגון גיד בין הגידים וביצה בין הבצים שההיתר לא קיבל טעם מן האיסור, וכיון דאיסורא לא מנכרא בטל ברוב, אבל כשקבל ההיתר טעם האיסור נעשה הכל איסור שהרי ניכר הוא וידוע בכל ההיתר וכו', אבל כל היתר שמקבל טעם האיסור וטעם האיסור ניכר בו, טעימתו זו היא הכרתו כדכתיב וחיך אוכל יטעם לו, אין חשיבות האוכל אלא בטעמו כו
When do we say 'one in two is nullified' according to the Torah? Only when the forbidden item is by itself and the permitted item is by itself... since the prohibited item is not discernible it is nullified by the majority. But when the permitted item acquires flavor from the prohibited item the entire thing becomes prohibited because it IS discernible and known throughout the entire permitted item...
The key point is that having one part nullified by the majority is only possible when the one part is not currently discernible. Since there is flavor, the whole concept falls apart.
The point of having 60:1 is generally just to guarantee there isn't any significant, discernible flavor, as is clear in YD 98:1 and other places.