Do adverbs only describe verbs?

Egypt and Tunisia have both taken steps to form a new government after the overthrow of Mubarak and Ben Ali respectively.

In this context, does respectively describe the steps that have been taken, or the overthrow of Mubarak and Ben Ali?

Respectively shows that Egypt has taken the steps mentioned after the overthrow of Mubarak, and that Tunisia has done so after the overthrow of Ben Ali.

As for your headline question, adverbs can modify verbs, adjectives and other adverbs, as well as a number of other word categories.

Adverbs do a lot of jobs. In this case though it is affecting the verbal phrase "taken steps to form a new government". It means that they have carried out that action in a relative manner.

Consider if we were to replace it with the adverb together. This would mean that they had acted with each other on the action "take steps to form a new government" (it would be an awkward but valid meaning). Or with the adverb rapidly, again it would affect that verbal phrase (and again it would be awkward and graceless to have the adverb so far from the verbal phrase without a good reason, but it would be valid).

More generally, adverbs can modify verbs, adjectives, other adverbs, entire clauses, entire sentences, and just about anything except determiners and nouns.