What about one of the Kendall's $\tau$s? They are a kind of rank correlation coefficient for ordinal data.
Here's an example with Stata and $\tau_{b}$. A value of $−1$ implies perfect negative association, and $+1$ indicates perfect agreement. Zero indicates the absence of association. Here we see a modest, though significant, negative association between speed limits and accidents.
. webuse hiway, clear
(Minnesota Highway Data, 1973)
. tab spdlimit rate, taub
| Accident rate per million
Speed | vehicle miles
Limit | Below 4 4-7 Above 7 | Total
-----------+---------------------------------+----------
40 | 1 0 0 | 1
45 | 1 1 1 | 3
50 | 1 4 2 | 7
55 | 10 4 1 | 15
60 | 9 2 0 | 11
65 | 1 0 0 | 1
70 | 1 0 0 | 1
-----------+---------------------------------+----------
Total | 24 11 4 | 39
Kendall's tau-b = -0.4026 ASE = 0.116
You can also try an asymmetric modification of $\tau_{b}$ that only corrects for ties of the independent variable. This is called Somer's D:
. somersd rate spdlimit
Somers' D with variable: rate
Transformation: Untransformed
Valid observations: 39
Symmetric 95% CI
------------------------------------------------------------------------------
| Jackknife
rate | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
spdlimit | -.4727723 .1395719 -3.39 0.001 -.7463282 -.1992163
------------------------------------------------------------------------------
All these measure of association are related in that they classify all pairs of observations (highways in our example) as concordant or discordant. A pair is concordant if the observation with the larger value of variable $X$ (speed limit) also has the larger value of variable $Y$ (accident rate). There are more of them than you can shake a stick at (one more is Goodman and Kruskal's $\gamma$, which ignores ties altogether like $\tau_{a}$). They will generally yield similar conclusions, even if they are not directly comparable.
The results above are qualitatively in line with Spearman's rank correlation coefficient mentioned by Greg (which tends to be larger in absolute value than $\tau$):
.ci2 rate spdlimit, spearman
Confidence interval for Spearman's rank correlation
of rate and spdlimit, based on Fisher's transformation.
Correlation = -0.451 on 39 observations (95% CI: -0.671 to -0.158)
This measure does not consider pairs, but compares the similarity of the ordering that you would get if you used each variable separately to rank observations (Stata breaks ties by assigning the average rank, and it's just Pearson correlation on the ranks). This makes it somewhat faster to compute since you don't have to consider all $\frac{n \cdot (n-1)}{2}$ pairs. On the other hand, the central limit theorem works much faster for $\tau$, so if you plan to do inference that measure might be better. $\tau_b$ is the most common variant.
Best Answer
A couple of caveats before moving on to the actual question you asked -
First, with 19 tests (really, more than one test) you should be adjusting for multiple comparisons. If you perform 20 independent tests of null hypotheses that are in fact true, you'd expect to get, on average, one p-value $\leq 0.05$, with the distinct possibility of getting more... which implies that your overall probability of rejecting a true null hypothesis is actually quite high.
One well-respected procedure you can use is the Benjamini-Hochberg procedure. You would need to choose a false discovery rate (FDR), that is, a desired maximum expected proportion of "discoveries" (i.e., rejections of the null hypothesis) that are false (i.e., the null hypothesis in those cases is actually true.) In your case, the criterion for your single "significant" correlation becomes $FDR/19$, which equals 0.0026 if you set the FDR at 0.05. You would conclude that you could not reject any of the null hypotheses at an FDR of 0.05.
Having written that, let's assume you set the FDR at 0.1 and consequently did reject the null hypothesis for the correlation above. Now, why is that correlation relatively large (in absolute value)? When you're comparing several (in this case) correlations, the largest ones probably got that way through some combination of underlying value and randomness, writing very loosely. This brings us to the second caveat - you can't trust that the largest effect size is an unbiased estimate of its true effect. There are ways of correcting for this, too, but I won't go into them here - they are more complex than FDR, and, depending on your data, may not make much difference.
On to the question! What does moderate correlation mean? I suspect you are overthinking the issue to some extent. It simply means that there is some relationship between the two variables in question, but that there's also a lot of randomness affecting one or both variables, or perhaps other variables affect the two variables in question, so the direct relationship isn't strong, but it's certainly noticeable. Plots help:
These two variables have a correlation of -0.44, about the same as yours. You can see there is a relationship, but it's certainly not a strong one, nor is it so weak it can be ignored. Hence, "moderate".