The coldest places are usually where it is winter. So in the Northern Hemisphere this tends to be January, and in the Southern Hemisphere it tends to be July.
Why is it not December and June? That is because the Earth, and in particular the oceans, soak up heat through the summer, and it takes time for that heat to be lost to space. So whereas the shortest day is in late December in the Northern Hemisphere, the coldest day is often in late January.
January and February are often nearly equally cold months even though, by February the Sun is rising higher in the sky. In very cold places there will be snow lying in these months, and the snow reflects the Sun’s heat.
The coldest place of all is not the North Pole. That is because the North Pole is surrounded by oceans. To get really cold you need to be away from the coast, and that means a relatively central place in a continent.
In winter the air over continents tends to sink as it gets cold, and this means no cloud forms and the sky is clear. This in turn allows the heat from the Earth to radiate out into space all day and all night.
The Antarctic is snowy all year, and it is a large continent, and so it is the coldest place on Earth. But that record only just beats Siberia in Russia.
The lowest natural temperature ever directly recorded at ground level on Earth was -89.2 °C (-128.6 °F), at the Soviet Vostok Station in Antarctica, on July 21, 1983.
Verkhoyansk, Siberia, is the coldest inhabited place on Earth. It recorded a temperature of -69.8 °C (-93.6 °F) for 3 consecutive nights, February 5, 6 and 7 in 1892.