Calvinism also asserts "irresistible grace" - it teaches that the atonement acts to effect salvation only for the elect - it overcomes any resistance, any rebellion, and it is an offer that they cannot refuse. Anyone whom God has elected to be saved, God will sovereignly bring them to salvation.
Arminianism counters with "resistible grace" - while Jesus' atonement in sufficient to save everyone, some people chose to resist it - they chose not to believe in God's offer of salvation. They resist God's grace.
One explains why only the elect will be saved - they can't resist God's grace.
The other explains that people have a choice and they can chose to not believe, they can choose to resist God's grace.
I remember one of the elders at my church putting it this way - God has his hand open, so that anyone willing to come in may, nobody can snatch anything out of his hand, but anything in his hand may jump out of it's own free will and God will respect it. Calvinism would teach that God snatched up the elect and is keeping them in his hand - their own wills are not free for them to do as they please. Nobody can say: "Hey, save me!" If they're not already snatched up by God because they're so lost that they couldn't if they wanted to. Nobody can say: "Hey, let me go!" Because God won't let them. I also remember being taught that God is a gentleman who won't go in where He's not invited and won't stay where He's not wanted and won't force himself on anyone. And yet, Calvinism points to a picture of Jesus knocking on a door: "Let me in!" "Why?" "So I can save you!" "Save me from what?" "From what I'm going to do to you if you don't let me in!"
I guess it all comes down to which picture is the best one of who God is - a gentleman, a pushy salesman that won't take 'no' for an answer, an open-handed individual who respects others, or a closed-fisted protector who only respects his own glory. Perhaps the better question is whether or not we really have a choice in anything - does God micromanage our salvation from eternity past, pre-selecting us as the elect, atoning for us, forcing grace upon us that we can't resist? Or Does God look down the road, see that we would chose to believe in him, atone for us, allowing us to chose grace and to choose to believe while allowing others to chose not to believe and not to accept his grace? Do we get the choice to choose one, the other, or neither of them?