The Sun releases huge amounts of energy through thermonuclear reactions in its core, where hydrogen is converted to helium. The energy created is in the form of electromagnetic radiation, effectively waves of electrical charge and magnetic force. This energy works its way to the surface of the Sun, heating the plasma (ionised gas - mostly hydrogen) that makes up the Sun. The Sun's surface, at 6,000 degrees, is much much cooler than the millions of degrees at the core.
From there, the radiation streams out into space at the speed of light, nearly 300,000 km per second. After about 8 minutes, some of it will intercept Earth. The Oceans and land surface will absorb some of the radiation and reflect some of it back into space. Radiation absorbed will heat the surface rocks, soil, ice and water. The land and seas in turn heat up the atmosphere through direct contact (conduction) and the heat is spread throughout the atmosphere through convection (air currents set in motion because warm air is lighter and will rise).
As the Earth's surface warms up, it radiates heat, mostly in the infrared part of the spectrum. When heat radiated equals incoming solar radiation, it stops heating up. When the sun sets, the Earth keeps on radiating and cooling until the next sunrise.