I suppose you have in mind the second part of 1 Tim 4:10: "Because we have our hope set on the living God, who is the Savior of all men, especially of those who believe." The interpretation of this verse has been problematic and consequently different explanations have been given to the words of the apostle. If you carefully read any English translation of this passage you will probably realize that it does not seem to make sense. How can everybody be saved and at the same time have a group that is specially saved? What does "specially" saved means? Does it mean that the others did not deserve to be saved and yet they are also going to be saved? When it comes to salvation one is saved or not saved. There is no middle ground; no one is going to be specially saved versus others who are only saved.
      Several solutions have been proposed to answer those questions. One of them argues that the word "Savior" means here "Benefactor" in the sense that God provides for the needs of all people. This idea is usually rejected because the term "Savior" in the Pastoral Epistles points to the salvific work of Christ in a very specific way. Another solution finds in this text the idea of universalism, that is to say, that God will ultimately save every human being. At the present time God is specially the Savior of believers but at the end He will show Himself to be the Savior of all people. No one will be lost. This teaching goes against the fact that according to the NT, some people will be eternally lost.
      The interpretation that you heard is another attempt to explain the passage. It argues that on the cross God legally saved the world (He is the "Savior of all men"), but that the individual is specially saved when that salvation is not rejected. There are several problems with this view. First, the idea that all men are already saved is not found in the Pastoral Epistles. For instance, in 1 Tim 2:4 Paul writes that God "desires all men to be saved and to come to the knowledge of the truth." If the interpretation under discussion were right one would expect Paul to say that God desires all men to recognize that they have been saved. But that is not what he says. Paul's statement indicates that it is incorrect to say that all humans are already saved in any form or way. Second, the idea of a legal salvation is not present in the text. It simply states that God is the Savior of all men and this is considered by the apostle to be a fact. Third, this view ignores that if "Savior" means that God already saved "all men," the term "specially" would still imply that "all men" were actually saved. That is to say, there is no indication in the text that salvation is given to some in a limited form, but fully to others.
      There is the possibility of understanding the phrase "the Savior of all men" on the basis of 1 Tim 2:4, that is to say, that God is the Savior of all in the sense that He desires "all men to be saved." Then, the phrase "specially of those who believe" would designate those who have accepted the offer of salvation by faith in Christ. In other words, God desires all to be saved and has done everything necessary to save them, but He is the actual Savior of those who believe. However, it appears to me there is a better solution.
      The key term in the text is the word "specially" (Greek málista). A proper understanding of it will solve the problems. The word can be translated "specially, mostly, above all" (e.g., Acts 20:38; Phil 4:22). Recent studies made on the usage of this term in Greek papyrus letters have shown that this particle was also used to define in a more precise way what preceded it and could be rendered into English as "that is, in other words" or by similar expressions. This usage is found in several passages in the NT (e.g., 2 Tim 4:13; Titus 1:10). If we use that translation in 1 Tim 4:10 the interpretational problems disappear: "Because we have our hope set on the living God, who is the Savior of all men, that is, of those who believe." The phrase "all men" would then mean "all sorts of people," possibly Jews and Gentiles, who accept Christ. There is no need to introduce in the text the idea of universalism or of a legal universal salvation which has no immediate effect on the final destiny of the individual.
      A few more comments. The text under consideration reads, "the Savior of all men;" it does not read, "He saved all men." We have a noun, "Savior," not a verb, "to save." There is a verb in the text, but the verb is in the present: ". . . God, who is the Savior of all men." The text is not describing what God did in the past but His constant activity toward humans. He wants all to be saved and He, through Christ, is the only one who can save all. The phrase "Savior of all/world" carries in the NT at least two important ideas. The first one is that the Savior did not come to save only his people, the Jews, but also the gentiles. This was clearly understood by the Samaritans who, motivated by the Samaritan woman, went to see Jesus and to hear his massage of salvation. At the end of the story they said to the woman, "We no longer believe just because of what you said; now we have heard for ourselves, and we know that this man really is the Savior of the world" (John 4:42). They realized that by offering them salvation Jesus had revealed himself not simply as the Savior of the Jews but as the Savior of the whole world. Salvation was not restricted to a particular ethnic group but was extended to all.
      Second, "Savior of all" also means that Jesus is the only and exclusive instrument of salvation for the human race. There is no other way accessible to human beings through which their relationship with God could be restored. No one needs to perish because he, as the only Savior available to all, offers salvation to all. Peter, in one of his sermons, stated, "Salvation is found in no one else, for there is not other name under heaven given to men by which we must be saved" (Acts 4:12). How, then, are we saved? Peter clarifies: "Repent and be baptized, every one of you, in the name of Jesus for the forgiveness of your sins" (3:38); "every one who believes in him receives forgiveness of sins through his name" (10:43). Peter does not know anything about universal legal justification. He knows that there is forgiveness available to all through Jesus and that we are forgiven when we believe.
      On the cross God made provision for the salvation of every human being but only those who through faith accept the gift of salvation through Christ will be saved. Christ's substitutionary atonement is universal in its extent but, because of the freedom God has given to humans, it is limited in its salvific effect.