Does the Bible say when the day of the Lord (as described in 2 Peter 3:10) will occur, and does it go into more detail about this particular event?

Andremonity has made the correct answer that every person with knowledge of the Bible would have.

32 “But about that day or hour no one knows, not even the angels in heaven, nor the Son, but only the Father. Mark 13

The Bible is very clear that no one knows, or even can know, when this event is going to happen.

Now, here is where I think Andremonity has made things a little too simplified.

A very large majority of Christians believe that this event is the Second Coming of Jesus Christ, which will be when all are judged by God and found worthy or not worthy to enter His Kingdom. The verse you quote is clear that it is going to be one heck of an event. Although I know of no Christian denomination that says it is a war, a good majority say that it ends a war (the battle of Armageddon).

It so happens that there are quite a few details about this exact event. There are details about what happens before and after it, too. They are all detailed in the New Testament's only prophetic book Revelation. The Second Coming of Christ starts at Revelation 19:11. Now, I should warn you that Revelation is the most argued about and most difficult to understand book in the entire Bible. I do encourage you to read through that little section and maybe chapter 20 too, but before you try to understand it, it would benefit you many times over to read and understand the rest of the New Testament first.

Does the Bible say when this event will occur, and does it go into more detail about this particular event?

2 Peter 3:10 But the day of the Lord will come as a thief in the night; in the which the heavens shall pass away with a great noise, and the elements shall melt with fervent heat, the earth also and the works that are therein shall be burned up

The Bible uses the phrase "the day of the Lord" frequently, and it can mean a specific event (such as the rapture), a series of events (such as the rapture and the Great Tribulation period), or a great span of events (such as everything from the rapture through to the millennial kingdom and to the eternal state).

1 Thessalonians 5:2-9 For yourselves know perfectly that the day of the Lord so cometh as a thief in the night. For when they shall say, Peace and safety; then sudden destruction cometh upon them, as travail upon a woman with child; and they shall not escape. But ye, brethren, are not in darkness, that that day should overtake you as a thief. Ye are all the children of light, and the children of the day: we are not of the night, nor of darkness. Therefore let us not sleep, as do others; but let us watch and be sober. For they that sleep sleep in the night; and they that be drunken are drunken in the night. But let us, who are of the day, be sober, putting on the breastplate of faith and love; and for an helmet, the hope of salvation. For God hath not appointed us to wrath, but to obtain salvation by our Lord Jesus Christ.

Zechariah 14:1-9 Behold, the day of the LORD cometh, and thy spoil shall be divided in the midst of thee. For I will gather all nations against Jerusalem to battle; and the city shall be taken, and the houses rifled, and the women ravished; and half of the city shall go forth into captivity, and the residue of the people shall not be cut off from the city. Then shall the LORD go forth, and fight against those nations, as when he fought in the day of battle. And his feet shall stand in that day upon the mount of Olives, which is before Jerusalem on the east, and the mount of Olives shall cleave in the midst thereof toward the east and toward the west, and there shall be a very great valley; and half of the mountain shall remove toward the north, and half of it toward the south. And ye shall flee to the valley of the mountains; for the valley of the mountains shall reach unto Azal: yea, ye shall flee, like as ye fled from before the earthquake in the days of Uzziah king of Judah: and the LORD my God shall come, and all the saints with thee. And it shall come to pass in that day, that the light shall not be clear, nor dark: But it shall be one day which shall be known to the LORD, not day, nor night: but it shall come to pass, that at evening time it shall be light. And it shall be in that day, that living waters shall go out from Jerusalem; half of them toward the former sea, and half of them toward the hinder sea: in summer and in winter shall it be. And the LORD shall be king over all the earth: in that day shall there be one LORD, and his name one.

Joel 3:14-21 Multitudes, multitudes in the valley of decision: for the day of the LORD is near in the valley of decision. The sun and the moon shall be darkened, and the stars shall withdraw their shining. The LORD also shall roar out of Zion, and utter his voice from Jerusalem; and the heavens and the earth shall shake: but the LORD will be the hope of his people, and the strength of the children of Israel. So shall ye know that I am the LORD your God dwelling in Zion, my holy mountain: then shall Jerusalem be holy, and there shall no strangers pass through her any more. And it shall come to pass in that day, that the mountains shall drop down new wine, and the hills shall flow with milk, and all the rivers of Judah shall flow with waters, and a fountain shall come forth of the house of the LORD, and shall water the valley of Shittim. Egypt shall be a desolation, and Edom shall be a desolate wilderness, for the violence against the children of Judah, because they have shed innocent blood in their land. But Judah shall dwell for ever, and Jerusalem from generation to generation. For I will cleanse their blood that I have not cleansed: for the LORD dwelleth in Zion.

I see in Peter's use of the phrase "the day of the Lord" a sort of a short parenthetical description of the entire period. The "thief in the night" phrase would describe the start of this time. The end of the old earth and heaven would describe the end of this time.

There is a parallel in the book of Revelation. After the first three chapters, the letter to the churches, the next two chapters describe a scene in heaven in which Jesus is found worthy to take ownership of the earth. The process of his taking ownership involves the opening of seals on the document that could be seen as a sort of title deed to earth. As this document is opened, various penalties are poured out on the earth. At the start of this process of judgment a war breaks out and together with famine and disease it causes the death of one-out-of-four people on earth. After the judgments are poured out on the inhabitants of earth, an angry army of 200,000,000 gather at Megiddo (an ancient Canaanite city southeast of Mount Carmel at the western approach of the Jezreel Valley in Israel) to fight against God.

Revelation 6:8 And I looked, and behold a pale horse: and his name that sat on him was Death, and Hell followed with him. And power was given unto them over the fourth part of the earth, to kill with sword, and with hunger, and with death, and with the beasts of the earth.

Revelation 16:16 And he gathered them together into a place called in the Hebrew tongue Armageddon.

After Jesus returns and wipes out the rebellious from the earth, he rules from Jerusalem for 1,000 years (i.e., the Millennial reign of Christ). After this, Satan, who had been bound, is set loose on the world, and he organizes one last war against God. After this war comes the Great White Throne judgment, where the unsaved dead are raised and judged according to their works. After this is the eternal state.

Revelation 21:1 And I saw a new heaven and a new earth: for the first heaven and the first earth were passed away; and there was no more sea.

T summarize, I suggest Peter is describing briefly the period of time (at least 1,007 years) which starts after the rapture.

2 Thessalonians 2:3-7 Let no man deceive you by any means: for that day shall not come, except there come a falling away first, and that man of sin be revealed, the son of perdition; Who opposeth and exalteth himself above all that is called God, or that is worshipped; so that he as God sitteth in the temple of God, shewing himself that he is God. Remember ye not, that, when I was yet with you, I told you these things? And now ye know what withholdeth that he might be revealed in his time. For the mystery of iniquity doth already work: only he who now letteth will let, until he be taken out of the way.

Since the time of the apostle Paul there has been someone restraining the evil in the world, and one possibility is the Holy Spirit, who indwells each Christian and is a restrainer of evil through each and every Christian throughout the world. When Christians are taken out of the world, the Holy Spirit would go with them. This might trigger all the events associated with "the day of the Lord."

1 Thessalonians 4:13-17 But I would not have you to be ignorant, brethren, concerning them which are asleep, that ye sorrow not, even as others which have no hope. For if we believe that Jesus died and rose again, even so them also which sleep in Jesus will God bring with him. For this we say unto you by the word of the Lord, that we which are alive and remain unto the coming of the Lord shall not prevent them which are asleep. For the Lord himself shall descend from heaven with a shout, with the voice of the archangel, and with the trump of God: and the dead in Christ shall rise first: Then we which are alive and remain shall be caught up together with them in the clouds, to meet the Lord in the air: and so shall we ever be with the Lord.

Here is the apostle Peter's summary concerning the transient nature of the earth and the heavens. Notice the similarity between it and the apostle John's summary:

2 Peter 3:11 Seeing then that all these things shall be dissolved, what manner of persons ought ye to be in all holy conversation and godliness,

1 John 2:16-17 For all that is in the world, the lust of the flesh, and the lust of the eyes, and the pride of life, is not of the Father, but is of the world. And the world passeth away, and the lust thereof: but he that doeth the will of God abideth for ever.

See Revelation 21:1. The Day of the Lord (the events of Revelation 6-20, Matthew 24, and elsewhere) will start unexpectedly, like a thief in the night. At the end of it, (Revelation 21:1), the heavens will pass away and the elements will melt, to be replaced by a new Heaven and Earth.