Let's start by reviewing the verses where that English word is used:
Rom 3:25 Whom God hath set forth to be a propitiation through faith in his blood, to declare his righteousness for the remission of sins that are past, through the forbearance of God;
1 John 2:2 And he is the propitiation for our sins: and not for ours only, but also for the sins of the whole world.
1 John 4:10 Herein is love, not that we loved God, but that he loved us, and sent his Son to be the propitiation for our sins.
Paul (in Romans) uses the noun ιλαστηριον, and John uses the noun (in two different declensions) ιλασμος (1 John 2:2) and ιλασμον (1 John 4:10). They are all related Greek words, and both are ordinarily translated by some form of the verb propitiate.
Webster's 1828 Dictionary defines propitiate:
PROPI''TIATE, v.t. [L. propitio; pio. Eng. pity.]
To conciliate; to appease one offended and render him favorable; to make propitious.
Let fierce Achilles, dreadful in his rage,
The god propitiate and the pest assuage.
Now, let's go back to the verses. In the verses, John and Paul are saying that Jesus is the one who appeases the offense that our sins cause to God and render us favorable in God's sight.
Now, in the verse Romans 3, I don't think anyone is going to have heartburn about what the verse says: it says that the Father put the Son in the role of being an appeaser of wrath by means of his blood to all those who believe: namely by a declaration of Christ's righteousness to remit the previous state of sinfulness, and, of course, through God's forbearance.
1 John 4:10 similarly does not cause heartburn, because John is writing from the apostles to believers, so whether "our" means the apostles or "our" means the apostles and the believers to whom he is writing, it is Jesus who assuaged God's wrath against us on account of our sins.
I John 2:2 is the verse that may cause Arminians some chest pains. It says that Jesus is not just the propitiation for "our" sins but for the sins of the "whole world." The problem for some Arminians is that they have adopted an exhaustive, universalistic sense (instead of a generic, expansive sense) to the term "world" in other passages, and feel obliged to understand the word similarly here, especially when accompanied by the word "whole."
Thus, Arminians are faced either with universal propitiation (and consequently universal salvation), or universal hypothetical propitiation. The problem with the former interpretation is that it is abundantly clear from Scripture that some will not be saved. The problem with latter interpretation is that there is nothing in the context to suggest that John or Paul believes that the propitiation of our sins is hypothetical or potential, as opposed to actual. Another option would be to suggest that the propitiation does not save, i.e. that propitiation is just an intermediate position between lost and saved. This, however, is clearly inconsistent with Paul's use, and there is - again - no reason in context to suggest this alternative.
There is, however, another solution to the dilemma:
The phrase translated "whole world" could be used in the generic, expansive sense - such that John is contrasting himself together with the apostles and/or his readers ("our") with other people generally. There is some support for this sense, because John uses the phrase in 1 John 5:19:
1 John 5:18-19
18We know that whosoever is born of God sinneth not; but he that is begotten of God keepeth himself, and that wicked one toucheth him not. 19And we know that we are of God, and the whole world lieth in wickedness.
Of course, here the "whole world" cannot be used in an exhaustive, universalistic sense - for John has just contrasted the "we" with it.
Furthermore, there are other passages where the term is used simply to convey a large amount (e.g. Matthew 16:26, Mark 8:36, Luke 9:25) or a broad expansive geographic area (Matthew 26: 13, Mark 14:9, Romans 1:8). In fact, in terms of Scriptural usage, those are all of the other (i.e. except for the two uses in 1 John) uses of the phrase. In other words, the phrase never in Scripture clearly conveys the sense of "each and every person who has or will live on the planet Earth," as Arminians are prone to think.
If we view John's comment as magnifying the greatness of the scope of God's love (i.e. that he propitiated for the sins not just of first century believers, but for the huge multitude of the elect), then the dilemma evaporates. There is no longer any need to eisegetically invent a hypothetical propitiation or a non-saving assuagement of wrath as one of the works of Christ.
And do you know what is really interesting?
The same Greek word ιλαστηριον is used by Paul (or whoever wrote Hebrews) to refer to the mercy seat, the cover of the ark of the testimony, the place where God communed with Moses, the places where God appeared in a cloud to Moses, and the place where blood was offered to God. It is the place where blood was offered to obtain mercy, and the author of Hebrews indicates that this blood sacrifice pictured the blood sacrifice of Christ.
May God who is Love, cause us to be thankful for the propitiation accomplished by our High Priest,