Two friends meeting
Friends often say "Hi" to each other. Then they often ask a general question, such as "How are you?" or "How are things?" or "How's life?"
The reply to this question is normally positive.
"Fine thanks, and you?"
"Fine thanks, what about yourself?"
"Not bad." Or "Can't complain."
Greeting people you don't know
You can use "Hello" with people you don't know, but a more formal greeting is "Good morning / afternoon / evening."
The other person normally replies with the same greeting as you have used and then makes polite conversation, such as "How was your trip?" or "Did you find our office easily?"
At an informal party
"Hello, I'm Maria." Or "Hello, my name's Maria."
The reply could be:
"Hi, I'm Sarah." Or "Hello Maria, I'm Sarah." Or "Nice to meet you, I'm Sarah."
At work-related events
"I'd like to introduce myself. I'm Maria, from english@home."
Or, "Let me introduce myself. I'm Maria from english@home."
The reply could be:
"Nice to meet you. I'm Peter Mitchell, from Mitchell Creations."
"Pleased to meet you. I'm Peter Mitchell, from Mitchell Creations."
"How do you do? I'm Peter Mitchell from Mitchell Creations."
Introducing other people
Introducing a friend to a work colleague
"Sarah, have you met my colleague John?"
"Sarah, I'd like you to meet my colleague John."
"Pleased to meet you, John." Or "Nice to meet you, John."
John could say:
"Nice to meet you too, Sarah." Or "Hello, Sarah."
"Mr Mitchell, I'd like to introduce you to my manager, Henry Lewis."
Mr Mitchell could then say:
"How do you do?" and Henry Lewis also says "How do you do?"
Or Mr Mitchell could say:
"Pleased to meet you." Or "Good to meet you."
"How do you do?" is quite formal for British English speakers and the reply to this question is to repeat the phrase, "How do you do?" (as strange as that may sound!)
At a more informal party
When you introduce two of your friends to each other, you can simply say, "John, this is Sarah."
At work, one person may have higher status – your boss, or a client, for example. It's polite to address them as Mr / Ms until the situation becomes more informal.
If someone says, "Please call me (Henry)", you know you can use first names. If someone uses your first name, you can use their first name too.
People in European and English-speaking cultures often shake hands when they meet someone for the first time.
* Don't forget to smile! :-)