In Calvinism, according to the idea of "limited atonement", only the sins of the Elect were atoned for in Jesus' death. While it would have been sufficient to have forgiven everyone's sin if it was God's plan to do so, it was only effective (able to actually save) to forgive the sins of the elect. Not a drop of his precious blood was spent needlessly.
Arminianism believes in "unlimited atonement" or "universal atonement", it states that the purpose of the atonement was universal and that Jesus died on behalf of all people, not just the elect. It says that the atonement makes a way for all to respond to the gospel so that salvation is available for all. It says that the atonement pays for the sins of those who would believe in Jesus.
I know that both belief systems (and probably a great many other ones) are rooted in Scripture, a verse here, a verse there - and look - they connect! But with both sides being found in the Bible, that means that you have a pro teaching and a con teaching in the same book. What do you do with verses that don't support your beliefs? Do you interpret them through the lens of what your beliefs tell you to be true? Do you just discount the veracity or worth of verses that don't agree with what you already believe? What is the good news - "that God so loved the world" in the light of that "He died to save only the elect and condemn the rest of the world to Hell?" Are the elect the whole of the "world" and the non-elect some "other world" that just doesn't count? To me, Calvinism doesn't offer even a ray of "hope" to those that it can't promise are saved. That's what worries me about it. We need hope - without it, getting through one day to the next gets extremely difficult when times get tough. It's the hope that we can be saved that helps us survive. But Calvinism erases that hope - sure, it's a sure thing for the elect - but for everyone else they're surely going to Hell and they have no hope of salvation.
I get it that they don't want to create a scenario where Jesus' blood atones everybody - believer or not. So that nobody is accountable for anything because Jesus forgives them without making them ask him to forgive them. But Calvinism doesn't seem to leave much room for people to chose to believe because if they're not elect - they can't. No matter how righteous a life they live, if they aren't elect they don't have a prayer of being saved because Jesus' blood doesn't work to save them. It would, If it were God's plan to save the whole world, but it's not - so they can't be saved. Why wouldn't God want to save everyone?