These two variables are almost uncorrelated. What else can I say? I have two variables for some districts in the UK:


*

*Number of crimes per habitant

*Median yearly income


Here's what it looks like:

I would like to determine if there is a dependence between the two. I ran a Pearson's correlation test, which gave me the following result:
    Pearson's product-moment correlation

data:  income and nbCrimesPerHab 
t = 1.3689, df = 315, p-value = 0.172
alternative hypothesis: true correlation is not equal to 0 
95 percent confidence interval:
 -0.03353993  0.18548945 
sample estimates:
      cor 
0.0769025 

As I understand it, the two are almost uncorrelated, but I am not sure how to interpret the p-value.
What could I do next? I am interested only in the crimes data. How could I determine their distribution? What more tools could I use?
EDIT:
As suggested in the comments, here's the same plot but this time with the mean

It doesn't seem to change much, but the correlation is a bit higer (and with a better p-value):
    Pearson's product-moment correlation

data:  income and nbCrimesPerHab 
t = 2.0986, df = 319, p-value = 0.03664
alternative hypothesis: true correlation is not equal to 0 
95 percent confidence interval:
 0.007318403 0.223310038 
sample estimates:
      cor 
0.1166938 

A scatter plot shows the structure better, as suggested:

 A: I think you need to consider two issues when interpreting correlations.
1) the p-value tells you the probability of observing an effect of this magnitude due to chance alone.  In your case, the probability that a correlation of this magnitude occurring solely by chance is 17.2%.  This is what it is.  Statistical convention typically dictates that we only accept as significant those tests in which this probability is < 5% (i.e., p-value <= 0.05) but I think it is better to consider the definition of the p-value as opposed to viewing 0.05 as a rigid wall.  
2) the second thing to consider is the correlation coefficient, which is the measure of how much of the variation in one random variable corresponds to the variation in another random variable.  This value can only range from 0 to 1 with a maximum at 1.
Based on your comment about being interested in the crimes data mainly, I think you may be more interested in regression since you are likely thinking about income as a fixed variable and are mostly interested in how crimes are dependent on income.  Your correlation coefficient is rather low (close to 0) so regardless of the p-value not much of the variation in crimes corresponds to the variation in income. 
Without knowing more about the questions of your study it is hard to suggest exactly where to go next.  The distribution of the crimes data can be visualized with a number of tools including boxplots, frequency histograms and scatter plots.    
Edit (based on comments)
Since you are interested in exploring other predictor variables you may want to consider stepwise multiple regression. This analysis allows you to build a multiple regression model from a suite of explanatory variables by only including those that improve the model by certain per-determined criteria.  Multiple regression will also allow you to assess the effect of one variable on crime after taking into account the effect of a previous variable.
Given your list of proposed variables (in the comment to @Leo) be sure to test for correlation among your explanatory variables (e.g., income and age) because the explanatory variables of a multiple regression (or even a series of independent regressions) should not be substantially correlated . If there is a lot of correlated between the predictor variables then you cannot effectively partition the influence of each predictor on the dependent variable.
A: So you don't want to add any other variables in your model (average age, level of education, ethnicity, etc.)?
It may be useful to visualize these two variables on a geographical map. You may see some patterns from there (including correlation, which may be slightly "shifted in space").
A: Pearson correlation assumes linear relationships. It is possible that you have a curvilinear relationship going on.That can often be the case when income is involved.  A second issue is that you have defined yearly income in terms of the median rather than the mean.  That may truly be hiding some relationships that you need to see. I suggest starting with a plot of the raw values prior to performing a correlation test to see if it reasonably meets the assumptions.
