Meet is usually a verb. Its past tense and -ed participle is met.
When you meet someone, you are in the same place and you start talking to each other.
I met a Swedish girl on the train.
I have never met his wife before.
When you intend to meet someone, you can say that you meet, meet with, or meet up with them.
This is an opportunity for parents to meet their child's teachers.
She's meeting up with some of her friends on Saturday to go shopping.
Meet with is especially common in American English.
We can meet with the professor Monday night.