Q: What does "mighty to save" mean?

The phrase comes from the book of Isaiah (63:1):

Who is this coming from Edom, 

       from Bozrah, with his garments stained crimson? 
       Who is this, robed in splendor, 
       striding forward in the greatness of his strength? 
       "It is I, speaking in righteousness, 
       mighty to save."

You may also be referring to the Hillsong song by the same name. Watch it below:

The chorus lyrics are as follows:

Savior
He can move the mountains
My God is mighty to save
He is mighty to save
Forever
Author of Salvation
He rose & conquered the grave
Jesus conquered the grave

I suppose the best way of answering this question is to examine the Scripture first, because - I assume - the song is based upon it. In this verse, Isaiah looks ahead to the future to see Edom and Bozrah - both unbelieving nations southeast of Jerusalem - being defeated by God. As all scripture is focused upon Jesus Christ, the text is ultimately describing the Messiah, returning from battling and conquering his enemies (Rev. 19:11–21). Edom and Bozrah, after all, typify those who disregard God. What this verse tell us is this: Though evil, injustice and suffering happen today, when Jesus returns, he will march "in the greatness of strength" (i.e. mightily) to save those who believe and repent, and defeat those that do not. 

This is what the famous preacher Charles H. Spurgeon said of this verse:

By the words “to save” we understand the whole of the great work of salvation, from the first holy desire onward to complete sanctification...Christ is not only “mighty to save” those who repent, but he is able to make men repent. He will carry those to heaven who believe; but he is, moreover, mighty to give men new hearts and to work faith in them. He is mighty to make the man who hates holiness love it, and to constrain the despiser of his name to bend the knee before him.

What Spurgeon is saying is that God's power is not only shown in the future day of judgment, but in the present: He has the greatness of strength to change a person's heart from rebellion & sin to salvation & holiness. 

My prayer is that God is mightily working in your heart today. 

[Answered by Pastor HM]