What does an observation essay look like and how should they be written when observing someone ?

You could start my addressing what it is you’re going to discuss in the essay and carry on from there with the structure of any other essay.
You need to focus on metacommunication – the communication within the communication. Body language, speech pattern, pauses, etc.
As he’s such a fantastic orator, you might want to discuss the techniques (for want of a better word) he uses.

There are some tips. I think keeping it in the present tense is probably the easiest to neglect.

Think in terms of what you learned in your speech class last semester about what being a good speaker, and use those to evaluate how the President performs as a public speaker.

Mr.Obama gave a speech to (fill in the blank) on (fill in the date) He entered the room to a crowd of (what kinds of people? Congress? Staff? A University audience? Reporters? Fill in the blank here). He was making this speech in response to (fill in the blank). Then at this point, make any observations about how the audience reacted to his entrance and beginning of his speech (were they standing, cheering, looking doubtful, sneering, smiling, nodding their heads in agreement etc.) Was the President or any of the audience members wearing any distinguishing pins or ribbons (like they did at the State of the Union address for Congresswoman Giffords) You can also mention what color tie he was wearing and if you noticed if his hair looks a little whiter or if he looked older or more tense than he did when he came into office.

Then give the highlights (with details) of the speech itself, and what kind of tone he used when delivering the speech (was it forceful? Light? Sympathetic? Angry? Resigned? or mixed depending upon what he was saying?) Use “keywords” to describe the topics. Use actual words that he used.

Talk about how long the speech lasted and what he said at the very end (in his conclusion). Then talk about how the audience reacted and what they all did immediately after the speech (Did they clap, Did they cheer, Did they look satisfied or irritated or upset?) and (did he simply leave the room? Did he stop to talk with anybody? Did he sign autographs? Did he walk away with his wife?)

You can also talk about any particular people that you noticed in the front seats of the audience, or anyone else in the audience that appeared to be singled out or was a guest of the President.

Then you can conclude with your “observations” about any themes that were brought up during the speech.

I hope this helps.

For an observation speech you probably need focus on a particular speech or speaking engagement, and not talk about his speaking style in general.