In any discussion of particular redemption it is of first importance that we define what is at issue. This is because many persons misperceive what Reformed persons actually affirm in the doctrine.
Here are some areas all evangelicals (including Reformed) have in common with regard to Christ's atonement.
1) Both Calvinists & non-Calvinists affirm that the redemptive benefits of the atonement will only be applied to a limited number of persons. The question is not, therefore, if the atonement is limited or not (this has already been determined) but rather, the question is who does the limiting? One believes it is "limited" by the will of fallen man and the other by the perfect will of God. It is not at issue (among evangelicals) as to whether all people will be saved. Christians of all stripes affirm that, when all is said and done, some persons will be redeemed by the blood of Christ and others will not; believers to eternal life and unbelievers to a resurrection of judgment where they will personally bear the full wrath of God for their sin.
Since all evangelicals believe some will be eternally lost, then it follows that the work of Christ on the cross does not bring to salvation the whole of humanity, and in this way all Christians, in some way, limit the atonement to a finite number of persons in the world. Those in the Reformed Tradition, therefore, do not limit the atonement any more than other evangelicals. Any person who denies universal salvation affirms some limits in the atonement. Therefore we believe "limited atonement" (perhaps a poor name choice for this beautiful doctrine) should preferably be called particular redemption - this describes God's revealed intent to redeem a particular people for Himself.
2) Particular Redemption is not a question about the value of Christ's work on the cross. Christ came and did exactly what he set out to accomplish. Those in the Reformed tradition affirm that the value of Christ's death is so vast that, in itself, is more than sufficient to cover all the sins of all the people in the whole world and a billion more worlds, if there were such a thing. In other words, Calvinists and non-Calvinists alike all believe that Christ's blood is of infinite value.
3) Calvinists and non-Calvinists alike believe there are some benefits in the atonement that are applicable to all persons, with redemptive benefits going only to some. Because of the work of Christ, Calvinists have traditionally affirmed that we must herald the gospel indiscriminately to all men, and proclaim that all who believe will enjoy the redemptive benefits of union with Him such as the forgiveness of sins and eternal life. Calvinists even affirm other benefits to non-believers, such as: that the work of Christ has prolonged the forbearance of God toward all of us who justly deserve His wrath. In other words, there has been a postponement of punishment and this because of Christ. Those who affirm particular redemption, do not affirm that Christ's work was only for the elect, but has universal implications for the gospel. However, Calvinists affirm that the redemptive benefits of Christ's crosswork is only intended for, and applied to the elect. Christ may love all persons in some ways but he only loves his bride in all ways.
So, then, what is at issue in particular redemption and why is it important? What is at issue is the intent of the atonement? What is it that the Holy Trinity had in mind in sending Christ to earth? Evangelicals agree that the Father sent the Son to be a vicarious substitutionary atonement. But for whom did Christ die? When he came to earth, which names did he have carved in his heart? Did he have the whole human race in mind, or was it those who were elect in Christ before the foundation of the world? (Eph 1: 4,5) Non-Calvinists will answer that it was for all humanity ... but Calvinists affirm that the redemptive benefits of Christ were only intended for those the Father has given the Son (John 17:9, 6:37, 39). Calvinists affirm that the Persons of the Trinity always act in harmony: The Father elects, the Eternal Son is sent to live and die for those the Father have given Him and the Holy Spirit applies the benefits to the same by bringing them into union with Christ.
It is in the grace of Christ itself that we even have the desire to pray or believe the gospel. No person will put faith in Christ who does not first have their heart changed by the Holy Spirit. The Apostle affirms that "no one can say 'Jesus is Lord' apart from the Holy Spirit" Effectual grace is not possible apart from the cross. The work of Christ is, therefore, effective toward all He intended to save. In other words, He fully accomplished what he set out to do. His effectual drawing of the elect is itself one of the benefits of the atonement. To separate this grace from Christ would be to affirm some kind of grace could be found outside the work of Christ, which is an impossible supposition -- for all spiritual blessings flow from Jesus Christ (Eph. 1:3).
Here are some passages for further study of the effectual work of Christ in the atonement
(Exodus 28:29, John 17:9, 20; Rom 8:34; Eph 5:25; Rev 5:9)
While all may not agree with conclusions reached by those who embrace particular redemption
my hope is that this has made some clarifications of what we believe.
Note: Four-point Calvinism fails the test of Christ-centered interpretation because this view tends to see the TULIP as an abstraction. But the TULIP only works when we see Christ at its center. Consider the TULIP as a chiasm with the "L" at the top of the pyramid. It is Jesus Christ which makes sense of all the doctrines of grace. Four-point Calvinists who reject particular redemption but embrace irresistible grace must consider this: Irresistible grace is not some abstract doctrine but must be seen in relation to Jesus Christ, specially in relation to the grace purchased by Christ upon the cross. The Spirit of Christ illuminates, regenerates and effectually brings to faith his elect. And this enabling, effectual grace is, from first to last, Christ-centered. It does not come out of a void, nor from some hidden source of grace in God the Father. Therefore Christ must have died for the elect so as to purchase that grace in a way - a redemptive way - that he did not die for the non-elect. That is why we often call it particular redemption. Irresistible grace is one of the redemptive benefits purchased by Jesus Christ ... and it was never granted to the non-elect nor intended for them. Jesus Christ is central to the TULIP and unless seen as the fountain of all blessings then irresistible grace will be seen as an Christless abstraction, rather than a benefit of our union with Christ.