Thank you for asking a question on www.thisisyourbible.com
The three "worlds" spoken of in 2 Peter 3 refer to three world "orders" existent upon the earth: one past, one present, and one future. The word translated "world" is the Greek word "Kosmos".
Here's how a Greek lexicon describes the word "Kosmos": an apt and harmonious arrangement or constitution, order, government; the inhabitants of the earth, men, the human family; the ungodly multitude; the whole mass of men alienated from God, and therefore hostile to the cause of Christ; world affairs, the aggregate of things earthly; the whole circle of earthly goods, endowments riches, advantages, pleasures, etc, which although hollow and frail and fleeting, stir desire, seduce from God and are obstacles to the cause of Christ; any aggregate or general collection of particulars of any sort....
2 Peter 3:5,6 speaks of the "kosmos" or world order which existed before the flood and was destroyed by the flood.
2 Peter 3:7 describes the "kosmos" or world order begun after the flood and which continues to our own day and is also reserved for destruction, this time by fire.
2 Peter 3:13 describes the "kosmos", "heavens and earth", which will constitute the righteous Kindom of God on earth.
That the righteous "New heavens and new earth" spoken of by Peter refers to an earth-based order of things can be seen from the Old Testament source of this phrase, Isaiah 65:17,18 For behold, I create new heavens and a new earth; And the former shall not be remembered or come to mind. But be glad and rejoice forever in what I create; For behold, I create Jerusalem as a rejoicing, And her people a joy.
Note that God declares that He will create a new "Heavens" and a new "Earth". What are these elements of the new divine "Kosmos"? He goes on to tell us; it is Jerusalem and her people! See Numbers 14:21 and Isaiah 2:1-4. That is the rulership of the Kingdom of God [Matthew 6:10; Daniel 2:44; 7:27; Revelation 5:9,10].
This also clarifies what Paul describes in 1 Corinthians 12:2 about his vision of being caught up to the "third heaven." This fits beautifully with what we have seen in 2 Peter 3:13. Paul has seen the "New Heaven" - the third "heaven" which rules the earthly "kosmos" - that is, the Kingdom of God which will have rulership over all the earth. What he saw was a vision of the Lord Jesus Christ and his saints ruling in glory in the "New Jerusalem."
Here is a link to some information on the website which will be useful to you:
The Kingdom of God on Earth www.thisisyourbible.com/library/show/170
The Kingdom of Heaven: www.thisisyourbible.com/library/show/1034
I hope you have found this helpful.
May God bless you,