The "day of wrath" is a phrase used in the Bible many times.
God is extremely merciful. In Matthew 5:44-45 we see Jesus say
"But I tell you, love your enemies! Pray for those who persecute you! 45 Then you will become children of your Father in heaven. For He makes His sun shine on good and bad people alike, and he sends rain to the righteous and the unrighteous alike" (my emphasis).
This is an important attribute of God. One which Jesus teaches us to imitate.
However just because God sends His rain and sun to the wicked doesn't mean He ignores their evil actions. Isaiah 57:6 reads
Your place (you wicked men) is among the smooth stones in the vadi;
these, these are what you deserve;
you pour out drink offerings to them,
you offer grain offerings to them.
Should I calmly ignore these things?
God is merciful and patient to the wicked, giving them time to repent. However the Bible shows us that the Proverb Jesus referenced when He tells us to do good to our enemies is for the purpose that doing good to them is meant to cause them Shame.
If thine enemy be hungry, give him bread to eat; and if he be thirsty, give him water to drink:
22 For thou shalt heap coals of fire upon his head, and the Lord shall reward thee (Proverbs 25:21).
"If your enemies are hungry, feed them. If they are thirsty, give them something to drink. In doing this, you will heap burning coals of shame on their heads" (Romans 12:20).
In other words, God isn't showing these acts of kindness for people to trample over Him and get away with their evil acts forever. The Prophet Elijah, one of the greatest prophets in the bible, was given power by God to stop the rain from falling for 3.5 years.
In the time of Noah God used the rain to kill the entire world except Noah and his family.
This shows there is an end to Gods patience, and when that end comes a horrible day of wrath begins.
The book of Psalms actually compares this to a drunken man awaking in blinding anger with no hopes to control it:
65 Then the Lord awoke, as if from sleep, like a warrior shouting because of wine. 66 He struck his foes, driving them back and putting them to perpetual shame (Psalm 78).
So to answer your question, when God has finally had enough with the evil ways of men on the Earth His great judgment will begin.
It is often that God begins by striking people's leaders, as with Pharaoh in Egypt. Before God judged all of Egypt He made an appearance to Pharaoh through Moses.
When people's leaders fall they often have no one to turn to but God, as was with many of the Egyptians who came to believe in God and in Moses. Some of the Egyptians even left with the children of Israel from Egypt (see here).
I hope that answers your question. If there's anything else you want to add or ask please leave a comment below.