A temperature gradient is the change in temperature across a specified boundary or front.
Why do I care? The temperature gradient between places results in differences in air pressure and ultimately, wind. Wind speeds increase with a greater temperature difference. Winds re-distribute energy around the world.
I should already be familiar with: What Drives Weather
Figure A. Temperature Difference Across a Frontal Boundary. (Image from University of Illinois at Urbana-Champaign).
A temperature gradient is just the change in temperature over a specified distance between two locations. The difference in temperature causes differences in air pressure between the two spots. This air pressure differential leads to the formation of winds as the atmosphere tries to equalize the air pressure. Generally, the larger the temperature difference, the stronger the resulting winds will be.
Winds around the globe are usually stronger in the winter, especially along fronts between highly contrasting air masses. There is a stronger temperature gradient between the poles and the equator during the winter months because the poles receive minimal sunlight at this time of year, while the tropics receive the same amount of solar energy year-round. This causes the jet stream to dip farther south and the winds to blow stronger on either side of the jet stream in winter. The jet stream separates the very cold air mass at higher latitudes from the warmer air mass equatorward.
Temperature gradients between water and land can also cause local atmospheric circulations which affect winds. During the day in summer when the land heats up more quickly than water, heat-related low pressure causes rising air over land which moves over the water and cools, then returns to land as a cooling "sea breeze". At night, the water is often warmer than the land and the reverse circulation, which includes a breeze from land to sea called a "land breeze", takes its place.