An example of a narrative speech would be a speech where a speaker talks about the difficulty that they had in assimilating in the U.S. school system when the speaker's family immigrated from Russia. Another example would be a speech where a speaker talks about the difficulty in losing his or her pet to terminal cancer.
A narrative speech is a speech that tells a story about a time in someone's life. It is a personal account and should be in narrative or story form. Narratives can also appear in other kinds of speeches to help make the speech more interesting and to support the points made in the speech.
The narrative speech is often used when the speaker needs to connect with an audience about their subject. Narrative speeches draw audiences in as they feel something in common with the speaker and can empathize with them. The story, however, should not feel forced and should not pander. A narrative speech that panders comes across as little more than begging and has a "feel sorry for me" tone that audiences are less likely to support.
The speech that detailed the speaker's difficulty in assimilating might be part of a speech for a charity event that works with immigrant children. The speech that detailed the speaker's difficulty in losing his pet might be for an organization that researches canine cancer.