Understanding the transformation on response variable Let me preface this by saying I'm new to statistics. 
I'm working with regression models, attempting to understand transformations a bit more. I'm modeling (Y~X) and I get an $R^2$ of 0.4. I see that the residuals of this plot are left skewed so I take (Y^2~X) assuming that would correct the issue but now my $R^2$ is 0.3. Just out of curiosity, I did (Log(Y)~X and got an $R^2$ of 0.5.
I'm really not sure what is going on and not sure what transformation I should use going forward. 
 A: The total sum of squares $\text{SST}=\sum(y_i-\bar y)^2$ will be altered by  transformation. 
The total variation available to be explained in the three cases ($Y_0=\log Y, Y_1=Y, Y_2=Y^2$) will be different. 
Specifically, if $Y$ tends to be substantially larger than $1$, you'll compress the variation by logging it and similarly expand the variation by squaring it (if $Y$ is positive but tends to be much smaller than $1$ then the log transform will stretch it and the square will compress it). 
That stretching/compression may tend to explain the changes in your $R^2$. 
A: From the reference I give below:  $R^2$ is explained as,
$R^2$ = $Explained \ Variation / Total \ Variation$   
where, 
1) $R^2$ is always between 0 and 100%:
2) 0% indicates that the model explains none of the variability of the response data around its mean.
3) 100% indicates that the model explains all the variability of the response data around its mean. the "variation divided by the total variation." 
Also from another reference:"...The coefficient of determination, $R^2$, is useful because it gives the proportion of the variance (fluctuation) of one variable that is predictable from the other variable. It is a measure that allows us to determine how certain one can be in making predictions from a certain model/graph."  In http://mathbits.com/MathBits/TISection/Statistics2/correlation.htm
Since expansion and contraction was covered by the other answer, I just want to make some comments on of aspects that effect the $R^2$.  An important part of $R^2$, is the selection of the function used to fit data.  You could have functions that would have have the same expansion or contraction and they can have different  $R^2$. With the results you mention and the $R^2$ given I would be incline to try a polynomial (in you independent variable) to see how it fits.  This function that contains multiple terms that are powers of x that would fit best, which would mean you can try a polynomial fit.  In the most general case of the polynomial, you would use spline regression to find an fit. The $R^2$ review is the first step in analyzing data.  Note: "Pearson Product-Moment Correlation" (which can be found on the Internet) discusses using $R^2$ to determine the "strength of the correlation."
The general reference on regression (and also "over fitting") I mention above a few times on how to interpret the correlation coefficient is "Regression Analysis: How Do I Interpret R-squared and Assess the Goodness-of-Fit?" is 
http://blog.minitab.com/blog/adventures-in-statistics/regression-analysis-how-do-i-interpret-r-squared-and-assess-the-goodness-of-fit.     
Lastly, just want to make a general comment on function selection vs model.  A model is development from the principles and laws of the field of study you are working in.  In many case a model is developed even before data is collected (e.g. Theoretical Physics -- I have done this many times).  On the other hand just selecting functions to try to "fit" data from experiments is not classified as a model. You are just looking for the best fit to the data (again as a first step) -- then others could study your data and develop/derive a model.   
