How can you tell if a yearly increase in population is statistically significant? I have daily data for two years consisting of "number of sightings" each day. Is there a way for me to test whether the data for year 2 is "significantly" higher than the first year?  
I know the average number per day for each year, as well as the standard deviation per year. The data, however, do not appear to be normally distributed.
 A: This isn't nearly as simple as you might hope. 
In particular, daily data within years are unlikely to be independent of each other.  Standard $t$-test routines take no account of that dependence, so the suggestion by @Behacad in a comment appears to be missing a major difficulty. You could do the calculation and your software may not complain, but the $P$-values and significance results could be way off.  
Whether data are normal is less crucial, for all that elementary texts are often obsessed with it. But depending on the properties of the counts, a Poisson or negative binomial or other discrete distribution may be more appropriate. 
In essence, you need a stochastic time series model for your data before you can formulate this problem properly, or minimally to use autocorrelations to correct the $t$- or other tests. 
On the other hand, plotting the data and doing a simple descriptive calculation may tell you as much about the data as you really need. If there is an underlying trend, then a year is an arbitrary time subdivision any way, regardless of it being well defined as a calendar unit. On the other hand, if "number of sightings" is something biological, years may be more natural units, especially if most of the activity is in a Northern Hemisphere summer, but then seasonality is likely to feature in the time series model you would ideally need. 
In essence, "statistical significance" does not cover all questions of the form "Should I regard this change as notable or big?". It only makes sense when a probability or stochastic model can be postulated for the data allowing the calculation to be carried out validly. 
Introductory texts or courses only rarely seem to comment on this common problem. Nor would changing the problem to something like Wilcoxon-Mann-Whitney be a solution, as independence is an assumption for those tests too. Box, Hunter, Hunter Statistics for experimenters from Wiley (either edition) and Rupert G. Miller Beyond ANOVA Wiley, reprinted by CRC Press are fine texts up-front about the problems caused by dependence. 
On the other hand, dealing with organism counts (if that is what this is) is surely a standard biological problem. This problem is likely to be covered in a methodological literature which is not very familiar to me. 
A: As @Nick Cox pointed out you could build a stochastic model for the full two years of data (730 values) given that there were not a lot of missing values or zeroes.  The model might include some ARIMA structure and/or possibly some deterministic structure to deal with  fixed effects such as day-pf-the-week effects (etc), perhaps monthly effects. I would definitely consider adjusting any of the 730 values for unusual activity i.e. 1 time pulses so as to make the subsequent analysis more robust. This can be done crudely or in a sophisticated way using Intervention Detection schemes which I prefer. Now armed with a reasonable XARMAX model where the X's are any needed deterministic variables , I would simply add a new variable: coded 365 0's and then 365 1's. Estimation of the final model would deliver a test of significance for this new variable which is a test of the hypothesis of equality of the two years. But that's just my take ...
One could use monthly values but with a count of 24 , I am not sure that this would always be appropriate but it would be worth a try. If the results don't prove anything one could always submit a paper to the "Journal of Negative Results".
A: I'm a Stata user so will defer to @Nick Cox.
If you had monthly data something like  interrupted time series +/- Prais-Winsten / Cochrane-Orcutt regression if you find significant autocorrelation would be useful. Daily data is usually very messy, and that approach might well break down.      
Aggregating data to month or even week would (1) throw away a huge amount of information and be hard to justify...but (2) likely give you an outcome that is easier to model in terms of distribution (distribution may well still be count data but with less zeros) and I would assume auto-correlation.
Standard review can be found here Wagner: Segmented Regression Analysis
I don't think this is what you're looking for but will give you more information: is there a secular trend, is is changing etc. 
