"Does an annulment say that my first marriage never existed?" No, a marriage took place - a civil marriage - but not the Sacrament of Marriage.
"Does an annulment mean that my children are illegitimate?" Absolutely not. Again, your children sprang from your marital union; but it wasn't a sacramental, supernatural, marital union. Even though the marriage ended in divorce, children demonstrate God's power to bring eternal blessings out of our imperfect choices. He used that imperfect marriage to bring forth new human beings to enrich the world and know and love Him for eternity!
"But didn't Jesus say that people could divorce if one of them had committed adultery?"
Great question. Let me copy and paste from The God Who is Love to answer this one.
There are Christian churches that allow divorce and remarriage, citing the parallel passage in Matthew :
And Pharisees came up to him and tested him by asking, “Is it lawful to divorce one’s wife for any cause?…Why then did Moses command one to give a certificate of divorce, and to put her away?”
[Jesus] said to them “For your hardness of heart Moses allowed you to divorce your wives, but from the beginning it was not so. And I say to you: whoever divorces his wife, except for unchastity, and marries another, commits adultery; and he who marries a divorced woman, commits adultery”
The disciples said to him, “If such is the case of a man with his wife, it is not expedient to marry.” But he said to them, “Not all men can receive this precept, but only those to whom it has been given…for there are eunuchs who have made themselves eunuchs for the sake of the kingdom of heaven. He who is able to receive this, let him receive it” (Matt.19:3,7-12).
Doesn’t this passage offer an exception to what seemed like the absolute prohibition on divorce in Mark’s Gospel? Wouldn’t an act of adultery, unchastity in this passage, break the marriage bond? I maintain that it does not – not when analyzed linguistically and contextually.
The term translated here as “unchastity” is porneia in Greek. At the time of Jesus one usage of the term was in reference to incestuous marriages. While such unions did not occur in Judaism (Leviticus 18:18-29), they did in the surrounding Gentile cultures – the very cultures that the Christian Church was expanding into when Mark and Matthew wrote. With these people’s acceptance of the Gospel would have come the recognition that their “marriages” were not valid before God. Matthew’s “exception” needs to be taken in context too; look at the conclusion the disciples drew from Jesus’ words, “If such is the case of a man with his wife, it is not expedient to marry.” The bond between husband and wife is so unalterable, and marriage therefore requires so much work, that the disciples concluded it was better to just never take the plunge! Jesus reminded them however that it is only by God’s grace, that a husband and wife can live out the marriage covenant. Now what sense would that exchange between the disciples and Jesus make if the marriage could be voided simply by one of the spouses cheating? It wouldn’t. No, Jesus’ prohibition of divorce and remarriage, recorded in Mark, was an absolute prohibition and our English translations of Matthew faulty.
“Porneia” and this issue of the invalidity of certain marriages does bring up another topic however, what the Catholic Church calls an annulment – the recognition that the Sacrament of Marriage never took place. Taking its cue from Jesus, the Church distinguishes between sacramental and legal marriages. Two Christians could profess vows to one another, but if one of them interiorly had no intention of remaining faithful or of being open to children, then God did not unite the two If one of the spouses were emotionally or physically coerced into marriage the same would be true. Without an investigation into the particulars, the Catholic Church operates under the assumption that a marriage between baptized Christians, Catholic or not, is sacramental. Before someone could “remarry” in the Catholic Church he/she would have to go through a process of showing why the first marriage was not a sacrament. The Church would then issue an annulment, a statement that the first marriage may have been binding legally, but not sacramentally.