What really makes RNNs powerful is the ability to operate over sequences of vectors, where both the input to an RNN and/or the output of an RNN can be a sequence. This is well represented by the following figure, where the leftmost example is a traditional (non-recursive) network, followed by an RNN with a sequence in output, followed by an RNN with a sequence in input, followed by an RNN with sequences both in input and in output where the sequences are not synced, followed by an RNN with sequences both in input and in output where the sequences are synced:

Machine translation is an example of non-synced sequences both in input and output: the network reads the input text as a sequence and after reading the full text, it outputs the target language.

Video classification is an example of synced sequences both in input and output: a video input is a sequence of frames and for each frame, a classification label is provided in output.

Andrej Karpathy's blog posting at http://karpathy.github.io/2015/05/21/rnn-effectiveness/ is a must-read if you want to know more about fun applications of RNNs. He trained networks to write essays in Shakespeare's style (in Karpathy's words: can barely recognize these samples from actual Shakespeare), for writing realistic Wikipedia articles on imaginary subjects, for writing realistic theorem proofs of silly and unrealistic problems (in Karpathy's words: More hallucinated algebraic geometry) , and to write realistic fragments of Linux code (in Karpathy's words: the model first recites the GNU license character by character, samples a few includes, generates some macros and then dives into the code).

The following example is taken from http://karpathy.github.io/2015/05/21/rnn-effectiveness/: