At any rate, at some point in these talks, the speaker uses the story of Cornelius as a counter-argument to the idea of Total Depravity. According to this speaker, Cornelius was an unregenerate man who feared God, and Heaven took notice of his piety(!).
At the root of his misapprehension is the idea that "salvation," "regeneration," and "conversion" are all synonyms. They are not.
William Kelly can explain this better than I, so I'll stop my own commentary here and quote his.
From Introductory Lectures on the Acts:
It is a fallacy then to suppose that Cornelius had no better than merely natural religion. He was assuredly, before Peter went, a converted man. To regard him as unawakened at that time is to mistake a great deal of the teaching of the chapter. Not that one would deny that a mighty work was then wrought in Cornelius. We must not limit, as ignorant people do, the operation of the Holy Spirit to the new birth. No man in his natural state could pray, nor serve God acceptably, as Cornelius did. One must be born again; but, like many others who had really been quickened in those days (and it may be even now, I presume), a soul might be born again, and yet far from resting in peace on redemption, far indeed from a sense of deliverance from all questions as to his soul. There is this difference, no doubt, between such cases now and that of Cornelius then, — that, before the mission of Peter, it would have been presumptuous for a Gentile to have pretended to salvation; now it is the fruit of unbelief for a believer to question it. A soul that now looks to Jesus ought to rest without question on redemption; but we must remember that at this time Jesus was not yet publicly preached to the Gentiles — not yet freely and fully proclaimed according to the riches of grace. Therefore, the more godly Cornelius was, the less would he dare to put forth his hand for the blessing before the Lord told him to stretch it out. He did what, I have no doubt, was the right thing. He was truly in earnest before God. As we are told here — and the Spirit delights to give such an account — "he was a devout man, and one that feared God with all his house, which gave much alms to the people, and prayed to God alway."
Such was the man to whom God was about to send the gospel by Peter. Thus we must carefully remember that the gospel brings more than conversion to God. It is the message of life, but it is also the means of peace. Before the gospel was preached to every creature, a new nature was communicated to many a soul; but till then there was not and could not be peace. The two things are both brought us in the gospel — life brought to light, and the peace preached that was made by the blood of the cross. At the same time scripture shows there might be and often was an interval after the gospel did go forth. So from experience we know there is many a man that you cannot doubt to be truly looking to the Lord, yet far from resting in the peace of God. Cornelius, I apprehend, was just in this case. He would no more have perished, had it pleased God to have taken him away in this state, than any Old Testament saint, whether Jew or Gentile. No believer could be so ignorant of God and His ways of old as to imagine there ought to be any doubt about those who nevertheless were full of anxieties and troubles, and through fear of death were all their lifetime subject to bondage.
And also from Kelly, from An Exposition of the Acts of the Apostles, Chapter 11:
Again, let us remark how clearly this discourse of Peter distinguishes new birth from salvation. Cornelius was assuredly born of God before Peter visited him at Caesarea. Nevertheless Peter was to speak to him words whereby he should be saved. It is a gross mistake to suppose that the salvation which he now found is not far beyond new birth. Present salvation is the first foundation privilege of the gospel. To be born again was always true from Abel downwards. But those who are merely born again do not enter Christian ground until they have received at least the first and most needful blessing, to which the accomplishment of Christ's work entitles all who believe...
The remarkable care with which God introduced the new standing-point [of salvation] to the Gentiles makes this confusion inexcusable. Now, while faith never was without suited mercy from God, it is one of the most marked signs of unbelief to ignore the peculiar privilege which God is now giving, and to go back to that mode or means which may have been at a former time. Here, as has been already and often pointed out, the Evangelicals are as dark as the Sacramentarians. For, if the latter party attach exorbitant efficacy to the mere sign of the blessing, the former are as ignorant of what is signified. Both agree in making the initiatory institution of the gospel to be the sign of life or the new birth; whereas it is really of the remission or washing away of sins (Acts 2:38; Acts 22:16), and of death with Christ (Rom. 6:2-3; Col. 2:12), i.e., of salvation (1 Peter 3:21). Cornelius learnt from the apostle that for a Gentile it was no question any longer of God's uncovenanted mercy. He himself, already born of God and acquainted with the Messiah come for the deliverance of His ancient people by faith, had now to learn of salvation's door open to the Gentile believer as truly as to the Jewish. It is not promise, as hitherto even to an Israelite, it is the work accomplished, and soul-salvation henceforth given to all believers without distinction. As the seal of it, the Holy Ghost was manifestly imparted as on the day of Pentecost.
It's worth taking a look at From New Birth to New Creation by R. A. Huebner, (especially pages 16–17) as well.
We can group Lydia (Acts 16:14ff) and John's disciples in Ephesus (Acts 19:1ff) in with Cornelius. These people were worshiping the one true God before they heard the Gospel. They weren't pagans, they were in the same position as Old Testament saints. For them the Gospel wasn't a call from death to life, but a call into the new order that God has begun with the death, burial, and resurrection of Christ.
Well, I promised that I would limit my own commentary and just use Kelly's, so I'll stop here.