This verse tells us there is a sense in which God is the Saviour of everybody, and also a special sense in which he is the Saviour of those who believe. It helps to understand this distinction if we realize that the word Saviour has the connotation of "Preserver" in a number of ways.
In the first sense, God preserves life on earth from being snuffed out. History records many miraculous interventions and providences that have made the difference at critical times. He brings restraints that keep the evil of man from destroying himself and others. It is because of God's preservation that we have not had a nuclear holocaust. There have been terrible plagues, but never one which destroyed mankind.
This preservation is active as well as preventative. "Nevertheless he left not himself without witness, in that he did good, and gave us rain from heaven, and fruitful seasons, filling our hearts with food and gladness" (Acts 14:17). In the final analysis, it is God that feeds us. "He maketh his sun to rise on the evil and on the good, and sendeth rain on the just and on the unjust" (Matthew 5:45b).
God's preservation is particular as well as general, for each one of us is alive today by the preserving hand God. Many will tell of remarkable escapes from death, sometimes as answers to their prayers. We should understand, however, that this general preservation is temporary, and nobody knows how long this life will continue.
There is a second kind of preservation. It is the provision God made for us spiritually and eternally through the saving work of His Son, the Lord Jesus Christ. This is the "special" sense in which God is the Saviour.
This preserves us from the wrath of God that was occasioned by our sins. By His forgiveness of sin, our greatest problem is solved and we are restored to God's favour. This preservation is distinctive also for its promise of eternity. "For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life" (John 3:16). It is only those who believe that can call themselves preserved (saved) in this full sense.