An extended passage of dialogue is, at its most basic, a large section which is made up of nothing more than characters talking to each other, without any interruptions from the author to give us descriptions or insights or other bits of information that might break the flow of the conversation we are hearing too as readers. This is a technique used by many authors to show us, in a subtle fashion, more about the characters involved in the dialogue and their motives and what drives them as individuals at that given time. Often the reader has to be attentive and alert to capture the nuances of such passages of extended dialogue.
One of the best examples of this method being used to devastating effect is in "Hills Like White Elephants" by Ernest Hemmingway, which is a short story which features the conversation of a woman and her lover where it is decided that she will get an abortion (against what the woman wants). One passage of extended dialogue is as follows:
"What did you say?"
"I said we could have everything."
"We can have everything."
"No, we can't."
"We can have the whole world."
"No, we can't."
"We can go everywhere."
"No, we can't. It isn't ours any more."
What is key to this conversation is the subtext - what is really being said and referred to, of which both the characters are painfully aware. It is a great story and well worth reading. I have included a link to information about it below. Enjoy!
I am not aware of specific rules for writing an extended passage of dialogue, but I expect it is the same as writing dialogue, except that there is no narration included.
For example, when reading a book, there is often dialogue mixed in with descriptions of setting, other characters, plot development, etc. I would expect that with an extended passage of dialogue, you would simply write dialogue without interrupting the flow of speaking. Open a novel and look for examples.
A passage indicates a section that is extended; it cannot be a short writing. To this end, the content of the passage should be well-developed with a specific purpose in mind.
If you haven't been given a prompt, I would imagine a situation that you feel compelled to discuss put into dialogue would come more easily, or perhaps a conversation you wish you could have with someone, but make sure the topic has substance, meaning. Provide both sides of the conversation, one speaker to another. Make sure to use quotation marks at the beginning of the passage, and after the end punctuation. For example, "Sally sells seashells at the seashore." (The period goes inside the end quotation mark.)
Avoid filler: like running on with words without clear intention. Your professor will see through it immediately. If you talk for extended periods of time on the phone or participate in debates with friends, etc., use one of these conversations as a model: writing from something important to you provides credibility to your writing.
In terms of conversational tags (he said/she said phrases), without specific instructions, I would avoid them. (she said, he explained, they pondered, etc.)
Make sure the passage has substance. Proofread for spelling, grammar and punctuation. Choose one tense and stick to it: all present or all past. No explanation/introduction of your passage should be needed if your content is purposeful.