Why does weather often affect mood? There are plenty of people who feel sad, sometimes even depressed when it rains or storms. This effect is often featured in writing and song, where the weather is often a sign of something sinister ("It was a dark and stormy night" is a very cliche beginning for darker stories). I'm curious as to why that is.

I do understand that there are certain activities that can't be done when it rains, but there are "rainy day activities" that take their place. What is it about the weather that makes people feel "down"?
The answer is that we don't really know (for non-clinical populations). The relationship between weather and mood isn't straightforward, and the impact of weather on mood may be very small (on average). 

There have been some great experience-sampling studies that have looked at this. For example, in a sample of 1233, Denissen, Butalid, Penke, and van Aken (2008) found that:


Daily weather had no significant effects on positive affect
Higher temperature was associated with higher negative affect
Lower wind power and lower sunlight were associated with higher negative affect
Lower sunlight was associated with greater tiredness
Being outdoors moderated the impact of some weather variables on affect
The impact of weather on mood varied significantly across individuals
The overall effect of weather on mood was very small


Kööts, Realo, and Allik (2011) found that:


Higher temperature was associated with increased positive and negative affect
More sunlight was associated with increased positive affect (but not with negative affect)
Temperature and luminance were inversely related to feelings of fatigue
Being outdoors interacted with weather variables to predict negative and positive affect
Neuroticism may account for some individuals' greater reactivity to weather


Keller et al. (2005) found that:


The interaction between more time spent outdoors and higher temperature and barometric pressure predicted greater positive affect. 
Interestingly: 


  Among participants who spent more than 30 min outside, higher temperature and pressure were associated with higher moods, but among those who spent 30 min or less outside, this relationship was reversed.
  
  Conclusion: As stated, the effects of weather on mood aren't linear. Sunlight can boost serotonin levels, which may explain why spending time outdoors improves mood (even when controlling for physical activity). This is corroborated by studies using light therapy for SAD or other forms of depression. And more obviously, we feel fatigued/tired as sunlight lessens. But overall, these effects aren't very big, and they vary quite a bit across individuals (indicating the effects of moderating variables like neuroticism).