If you know two people who haven't met, you can introduce them to each other. If you want to meet someone, you can introduce yourself. Here are some examples of phrases of introduction.
When Introducing Other People (starting with formal versions and moving to informal versions)
When Introducing Yourself (starting with formal versions and moving to informal versions)
Greeting Responses that Follow an Introduction (starting with formal versions and moving to informal versions)
Greeting
Response
Greeting Responses for People You Know (starting with formal versions and moving to informal versions)
Greeting
Response
Jazz Chant "Harry, this is Mary"
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-Harry, this is Mary!
-Hi!
-Hello!
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-Harry, this is Mary!
-How do you do!
-How do you do!
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-Harry, I'd like you to meet my sister Mary.
-How do you do, Mary. I'm very glad to meet you!
-Thank you. I'm glad to meet you.