If you are putting together an introductory speech, the first thing you have to do is figure out what the rhetorical situation is, or what the basic reasons are for giving the speech. Are you introducing yourself, another person who will also be speaking, or someone who is not even in attendance at the speaking engagement? Being assigned an introductory speech for class probably means you are either introducing yourself or describing a famous person who is not in your class, while giving an introductory speech at a special occasion like an awards ceremony or a banquet probably means you are introducing someone else in attendance. And without at least a modicum of training, something like an athletic banquet introduction speech can be stressful affair. Which means you should give it a little practice first.
Know what type of introductory speech you will be giving? Good. Now let's briefly take a look at each, starting with...
Unless you are briefly stating your credentials at the beginning of a speech, there are very few real world instances where you would introduce yourself, and certainly none where the entire point of the speech is to introduce yourself. So I'm just going to assume if you're doing this type of introductory speech, you were assigned to do it for a public speaking class.
That being said, you still want to make this type of speech as interesting as possible without coming across as arrogant. One good way to do this is to tell a story about your own life that focuses more on an important role model rather than yourself. Describe how a family member or friend taught you an important lesson that you always remember, or how they started you on the path of interest in a particular hobby. That way, your speech still helps to introduce you, but it also shows that you know you couldn't have accomplished everything in your life without the help of others.
Ultimately, introducing yourself in a speech should also serve as a way to introduce someone else. But if introducing someone else is already the reason for your speech, even better! That means it's time to look at our next type of introductory speech...
Introducing Someone Else (Who Is Not There)
This type of introductory speech is a much better assignment than introducing yourself. There is no risk of coming across as arrogant or too self-involved, and if the person you are introducing isn't there, there's very little pressure to speak of them in a certain way or sugarcoat the facts. Certainly don't slander them, but don't be afraid to let your audience know what kind of person they really are, either. Celebrities or historical figures are great subjects for this type of introductory speech, as your audience will likely know who they are and be interested to hear more information about them.
To hold your audience's attention, use a strong organizational pattern for your speech that focuses on one clear thesis statement about the person you are introducing. Then, back up your thesis with a series of main points (3-5) and supporting details with cited sources for any facts you give. Again, it's important not to slander your subject or misinform your audience, so don't spread baseless rumors, no matter how interesting they might be.
Introducing a famous figure in a speech is a good challenge for a classroom environment, but if you continue with a career that uses public speaking, eventually you will be...
Introducing Someone Else (Who Is There)
Outside of a classroom, speeches where you introduce someone else present are typically the types of introductory speeches you will most often have to give, assuming you ever have to give any at all. Maybe after you finish school, your life will be totally devoid of giving introductory speeches. Unless you decide to give an impromptu toast to Uncle Bob at the next family BBQ. It could happen!
Anyway, these types of introductory speeches depend a lot on the setting and event. Some common environments for giving introductory speeches include:
1) Weddings - Best Man toasting the happy couple. If you are giving this type of introductory speech, leave out any mention of strippers.
2) Awards Ceremonies - Presenting the award to the surely deserving recipient.
3) Conventions - Keynote speakers and panel moderators. These usually require a balance of professionalism and humor.
4) Roasts - Take those celebrities down a peg or two. The more mentions of strippers, the better.
5) Funerals - Eulogy given to the deceased (who is still there, just not there).
Like the other types of introductory speeches, introducing someone who is present requires a deft touch with humor and strong organizational skills. Since the person you are introducing will likely be speaking after you, keep it brief but memorable.
In conclusion, introductory speeches are a good way to help your audience become more familiar with a celebrity, important figure, or even yourself. Keep your sense of humor about you and deliver a clear and dynamic speech, and you will be sure to win the approval of your audience.