The December Solstice is the winter solstice in the Northern Hemisphere, while in the Southern Hemisphere it's the June Solstice. (Not to scale)
Solstices are opposite on either side of the equator, so the winter solstice in the Northern Hemisphere is the summer solstice in the Southern Hemisphere and vice versa.
In the USA and some other areas in the Northern Hemisphere, the winter solstice marks the first day of winter. However, the official date for the first day of winter varies depending on the country's climate, and whether they follow astronomical or meteorological seasons.
Northern Hemisphere Winter Solstice: (North America, Central America, Europe, Asia, northern Africa)
Southern Hemisphere Winter Solstice: (Australia, New Zealand, South America, Southern Africa)