The atmosphere responds to solar variability on three different timescales.
We can examine this variability using atmospheric models. The "recurrent"
effects due to diurnal, seasonal and base solar cycle are easiest to explain 
and illustrate - though even there we are finding in recent years that there
are things we don't understand. The next order of complexity is the response
to solar storms and other solar events. We can understand many of the gross 
features of the atmospheric response in ionosphere and thermosphere, but it 
is a highly complex system and, like ground-based meteorology, our ability to, 
"now-cast" it relies on how much input data we have. We don't have enough,
though we probably understand the underlying physics. What is mos contentious
is how much the effects penetrate lower in altitude and latitude into the
atmosphere. We show modelling results that suggest Nitric oxide transport from
major events could damage the low-latitude Ozone layer. The least understood
solar-terrestrial variability connection is the climate question. Correlation,
studies suggest a correlation exists between solar cycle(s) and ground-based
weather/climate, but it is difficult to see what the mechanism for this might 
be. A few suggestions have been made, but the evidence for all of them is hard
to come by.