A phrase substituted for another that would be considered offensive, blasphemous, or otherwise unsuitable to use. Sometimes made up at the last moment just before the exclamation, minced oaths tend to sound very similar to the oath or curse they are used in place of. Examples of this include substituting "screw you" for "fuck you" or "for Pete's sake" for "for God's sake" in the presence of people who take offense to hearing the latter curses (or children, whose parents typically don't want their kids exposed to words with harsh connotations until later in life). The term minced oath comes from the word mince, of which one meaning is "to suppress or weaken the force of" (Webster 1913), and the fact that these were oaths (or swears or curses).
The difference between a minced oath and a euphemism isn't particularly big: A minced oath is a euphemistic phrase. A euphemism isn't necessarily a phrase though. Saying someone passed on instead of died isn't a minced oath as the word(s) being replaced aren't any type of oath or curse but just another term. All minced oaths are euphemisms but not all euphemisms are minced oaths.
Some strongly religious people claim that use a minced oath in the place of something ordinarily considered blasphemous, such as saying "God damn it!" when stubbing one's toe, which goes against a certain religious commandment of not taking God's name in vain is still blasphemous as the minced oath is merely a euphemism for the same thing. So even if someone says "Gosh darn it!" instead of "God damn it!," some still consider the exclamation blasphemous. I'm not sure what these people would think of the minced oath "for Pete's sake," as it is generally thought to refer to Saint Peter and certainly not God. On the other hand, I've encountered many religious people who are apparently fine with using "Gosh darn" in the place of "God damn" and the like.
Minced oaths aren't solely the product of trying to get around religious commandments but that and trying not to swear in front of children (or influence them to swear) seem to be the largest reasons for the creation and usage of them.