Introducing other people does not have to be scary. I always tell my students to keep it simple. You can use "this is" in every situation.
Bob, this is Carol. Carol, this is Bob.
Mr. Washington, this is Mr. Lincoln. Mr. Lincoln, this is Mr. Washington.
Sometimes, increased formality is desired. In those situations, you can use the following phrases.
"Mr. Janus, please allow me to introduce Ronald King."
"Emir Simpson, I would like you to meet my father, Robert Manning."
"Dr. Waters, I would like to introduce you to Ms. Carter."
People often include how they know the person being introduced.
"Mr. Garcia, this is Mr. Bhakta. He is the senior vice president of our marketing department. Mr. Bhakta, this is Mr. Garcia. He is thinking about switching his business to our firm."
"Mr. Baker, this is my wife Janice. Janice, this is my boss Charles Baker."
Some casual/less formal ways to introduce other people:
"I don't think you know Ally."
"Have you met Rita?"
"John, do you know Sara?"
The simplest response to give is, nice to meet you (Or, nice to meet you, too depending on who said it first.) Some simply say, hi.
Formal responses include:
"It's a pleasure to meet you."
"It's a pleasure to make your acquaintance."
"How do you do."
"Pleased to meet you."