How does the sun's atmosphere affect the planets?
Actually, our Sun doesn't really have an atmosphere in the sense we know it and apply to other celestial objects.
The sections of the Sun from outermost to the core are as follows: the photosphere, the region at the Sun's surface where virtually all visible light is emitted, the convection zone, where the Sun's matter is in constant convective motion, the radiation zone, where the energy travels outward in the form of electromagnetic radiation and the core, where nuclear reactions generate energy.
Above the photosphere lies the chromosphere, the Sun's lower "atmosphere". In the transition zone above the chromosphere, the temperature increases from a few thousand to around a million kelvins. Above the transition zone is the Sun's thin, hot upper atmosphere, the solar corona. At the distance of about 15 solar radii (away from the core), the gas in the corona is hot enough to escape the Sun's gravity and the corona begins to flow outward as solar wind.
The solar wind is what affects the planets. Occasional coronal mass ejections are huge blobs of magnetized gas escaping into inter-planetary space. Most of the solar wind flows outward from low-density regions of the corona called coronal holes.
This solar wind is highly ionised and poses a great danger to bodies without a stable magnetic field (from a stable, rotating ferromagnetic core) as the solar wind will completely destroy (ionise, in the sense where life cannot exist) any good chance of life forming.
Planetary atmospheres are also affected by the solar wind and that is a matter for another day because I think this answer is seriously long enough.