There are a number of ways of making a self introduction. It usually needs to be on a case-by-case basis; however, here's some basic advice and sample dialogues to get someone's attention and finally make an acquaintance.

Direct Introduction

The direct approach works for most people who have the confidence to do so. Simply go up to the person whom you want to introduce yourself to. Say "hello,' offer a handshake, and tell them your name.

A: Hi. I'm Peter Holden. I'm from Accounts. How do you do? (shake hands)
B: Hello. Nice to meet you, Peter. I'm Jason Warrick.

If, however, it is a group you are approaching, politely ask if you can join them.

A: Hello. Is it alright if I join you?
B: Sure, no problem.
A: My name's Gordon Brand. I'm new here.
B: Pleased to meet you Gordon. I'm Tom Bevan. (shakes hands)
A: Pleased to meet you, Tom.
C: And you too, Gordon.
B: And this is Benjamin Pratt.
A: Pleased to meet you, Benjamin. (shakes hands)

Giving a Compliment

Giving a compliment is also a good tactic. Remember to give a compliment that you really mean. Sincerity is the key here. You can start the conversation with a statement like, "I like your shirt' or "You have a nice watch'. The other party can reply with a "Thank you'. From that point, be prepared to talk about the object you are complimenting on to prove that you really admire it. After a minute or two, or when appropriate, start introducing yourself.

A: I love your shoes. Where did you get them?
B: Oh. Thanks. I got them from Harvey's just yesterday.
A: What kind of leather are they made of? It's got a really nice pattern.
B: Aligator, actually.
A: I don't dare to ask you how much they cost. But they look so expensive.
B: Not as much as you'd think. I got them in a closing down sale.
A: Really? I'm Francesca Tomlins by the way.
B: I'm Robert Downing. Pleased to meet you, Francesca.
A: Pleased to meet you too, Robert. (shakes hands)