What does it mean that God is the saviour of all people, especially those who believe?

Jesus saves us from two kinds of death. All will be resurrected and overcome mortality, becoming immortal, thus triumphing over "Physical" death. But only those who believe on Christ and accept his atonement will overcome "Spiritual" death (separation from God) and be allowed to dwell with God, attaining eternal life.

And all will have the opportunity to hear the Gospel, whether in the body or in the spirit. We see in 1Peter 3:18-20:

18 For Christ also has once suffered for sins, the just for the unjust, that he might bring us to God, being put to death in the flesh, but quickened by the Spirit: 

19 By which also he went and preached to the spirits in prison; 

20 Which sometime were disobedient, when once the long-suffering of God waited in the days of Noah, while the ark was a preparing, wherein few, that is, eight souls were saved by water

So we know that Jesus went and preached to "disobedient spirits" who had died in the time of Noah (He did this in the time between His crucifixion and His resurrection.) This means that those who died before Jesus came, who did not have the Gospel, will have an opportunity to choose salvation. Why else would he "preach" to spirits of the dead?

When Jesus said he wanted "all" to be saved (as mentioned by another answerer) He meant ALL. Not just those who heard His voice while He walked this earth, or those who heard it preached thereafter, but all souls who have ever lived. This includes those living and dying today in places where the Gospel is not yet preached.

Isn't God a Savior only to those who believe? What could Paul be getting at here?

This touches on a doctrine called "limited atonement" suggesting that Jesus died only for the sins of the elect. There are verses that seem not to support this view;

1 John 2:2 And he is the propitiation for our sins: and not for ours only, but also for the sins of the whole world.

If one considers a view of what might be called "unlimited atonment", it would not mean that everyone goes to heaven, but that the sins of everyone had been paid so that everyone has the potential to go to heaven.

God predestined some to have the grace necessary to accept what is offered to all.

Romans 8:29-30 For whom he did foreknow, he also did predestinate to be conformed to the image of his Son, that he might be the firstborn among many brethren. Moreover whom he did predestinate, them he also called: and whom he called, them he also justified: and whom he justified, them he also glorified.

It really isn't that confusing; What Paul is indicating is that God is the Savior of all people including those who came before Jesus, but he readily saves those who believe that Jesus Propitiated our sins.

Since millions of people lived and died before Jesus descended from Heaven and paid for the sins of all mankind God is also their Savior based on the sacrifice of his son. That saving grace extends to the past as well as to the future.

I think Paul is saying that Jesus is the saviour of all people. I don't think that the word "all" needs to be qualified in any way.

There are other "all" verses which help support this. For example:

• 1 John 2: 2 "He is the propitiation for our sins, and not for ours only but also for the sins of the whole world."

• Rom 5:18 "Consequently, just as one trespass resulted in condemnation for all people, so also one righteous act resulted in justification and life for all people."

• 1 Cor 15:28 "When he has done this, then the Son himself will be made subject to him who put everything under him, so that God may be All in all."

• Col 1:19 "For God was pleased to have all His fullness dwell in Him, 20 and through Him to reconcile to Himself all things, whether things on earth or things in heaven, by making peace through His blood, shed on the cross."

I think that God wants all to be saved (1 Tim 2: 10) and is able to achieve it and eventually will do so.