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Potential gradient is the negative of the electric field:

$dV=-\vec{E}\cdot \operatorname{d}\!\vec{r}$

Does the negative sign mean that the direction of potential gradient $\operatorname{d}\!V\!/\!\operatorname{d}\!\vec{r}$ is opposite to that of the electric field $\vec{E}$? And if it does mean so, why is its direction opposite to the field? Explanations would be helpful.

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One thing necessary about Potential. It's simply the work done in moving an electric charge against the electrical force. Thus, The negative sign actually says that the work done is against the electric force (either attraction or repulsion). Or in other words, electric potential decreases in the direction of electric field. Hence, you're quite right about the concept.

This link is explains in an imagery way.

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