How is it possible to have a significant correlation between two variables but a low covariance? How is it possible to have a significant correlation between two variables but a low covariance?  What does this mean?
 A: Welcome to site, @Debbie! First of all, if this is a homework question, please revise it and add the "self-study" tag.
The answer is, covariance equals the product of correlation and the standard deviations, i.e.,
$$\mathrm{cov}(X,Y)=\mathrm{cor}(X,Y)\mathrm{sd}(X)\mathrm{sd}(Y).$$
As shown in the small simulation below, when the standard deviations are small (0.1 and .1*$\sqrt{2}$ in the example), even the correlation is high and significant (cor = 0.68, p-value = 4.4e-15), the covariance can be low (0.008 here).
> set.seed(1)
> 
> x = rnorm(100,sd=.1)
> y = x + rnorm(100,sd=.1)
> 
> cor.test(x,y)

    Pearson's product-moment correlation

data:  x and y
t = 9.2729, df = 98, p-value = 4.441e-15
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 0.5627820 0.7758677
sample estimates:
      cor 
0.6836311 

> cov(x,y)
[1] 0.008059066

A: In short, anything can be significant if $n$ is large enough.
When estimating and significance testing the correlation coefficient $r$, we most often use the estimator $s$:
\begin{equation}
s=\frac{s_{xy}}{s_xs_y}
\end{equation}
Where $s_{xy}$ denotes the sample covariance and $s_x$ , $s_y$ the sample standard deviations. The standard error of this estimator is given by:
\begin{equation}
se_s=\sqrt{\frac{1-s^2}{n-2}}
\end{equation}
Significance testing is done by the estimator $t=\frac{s}{se_s}$ which is approximately t-distributed with $n-2$ degrees of freedom.
As $n$ becomes large, two things happen. Our estimator $t$ comes closer and closer to a  $N(0,1)$ distribution, and the standard error gets smaller. In a $N(0,1)$ distribution, we would reject the hypothesis $H_0: r = 0$ at the 5% level, claiming it significant, if $|t|>1.96$. 
But then, for any given variance and covariance, and thus any given $s$, a large enough $n$ will make this happen.
