Compliments are a very important part in social relationships. Paying compliments has a two-way reward: they feel better and you get more appreciation from them (which may be just nice or even useful). Here is a list of the compliments from this conversation and their replies:

- Well done on clinching that deal!
- Oh, thanks.

- You did really well.
- Thank you.

- You're a great manager, you know.
- Do you think so?
- Oh yeah.
- I appreciate that.

- They [at the hairdresser's] did a great job [your hair looks really nice] and it looks great with the outfit.
- Do you think so?
- Oh yea. Very... fashionable.

- You've got a great sense of style.
- Thank you, you too.

- Nice perfume [...] You wear it well.
- I'm flattered  (= I'm pleased with your compliment)

As you could see above, when somebody pays you a compliment the usual and appropriate reaction is simply to thank them. You can find in the conversation some ways of thanking them for the compliment:
- Oh, thanks
- Thank you
- Do you think so? (it's not really a question, just a way to get the other person say "oh, yea", which confirms their compliment)
- I appreciate that
- I'm flattered.
We can also return the compliment easily saying (if appropriate):
- Thank you, you too.

Rejecting a compliment (specially if it's the first one) will often make the other person feel stupid and embarrassed. Imagine this conversation:
- Nice sweater, it really has style.
- Oh, this? Oh no, it's just a cheap thing I bought in the sales.

In many cultures this would be the right reaction because it makes you look modest and humble instead of vain, but in Anglo-Saxon cultures, this false modesty would simply make the other person look stupid and with no sense of style. You're actually telling them: my sweater is horrible so you have a very poor taste for clothes, you can't tell a good sweater from a shabby one and either you have no sense of style, or you are simply lying to me because you think I am stupid and I'm going to believe anything you say.

If the other person was trying to flatter you just to be nice and start a conversation, after your "virtuously modest" reply, chances are that they're going to move away from you and look for somebody nicer.

You have to be careful with modesty if you don't want to create misunderstandings. If you are not very familiar with the people and the culture around you, it is safer to simply smile and say thanks. Nevertheless, sometimes we can show some modesty (especially if they pay exaggerated compliments or too many of them). Or maybe our modesty is simply looking for confirmation, forcing the other person to reaffirm their statement.

- Oh thanks. It wasn't too difficult.
- You're being modest! You did really well.
- Thank you.

- ... it looks great with the outfit.
- Do you think so?
- Oh yea. Very... fashionable.
- Oh, it's just an old thing.
- Well, you've got a great sense of style (if he didn't say anything here, he would be confirming your words: yes, you're right, it's an old thing and it looks old and horrible).

Another important thing you must bear in mind is that women and men react to compliments in different ways, so I'm going to make some comments about it and if you think they sound sexist is simply because they are (inasmuch as I believe men and women are psychologically different). This difference is similar in most cultures, so I don't think you will have problems understanding this:

Flattering a woman is usually quite easy, simply tell them nice things about them. Most of them will appreciate the compliment even if it's not true, because they appreciate the good intentions behind your words. So if you tell them "you look great", they'll appreciate that, even if they know that they had no sleep and look horrible. A compliment for most women is like a flower, it's always nice to get one. But complimenting a man is not so easy, some compliments are well accepted (especially if they praise their vanity, intelligence, worth or masculinity) and some just make them feel uncomfortable (for example praising their good looks and appearance may make them feel good or embarrased or even silly, and much more so if the praise comes from another man).

But in either case, be very careful, in most situations, if you praise someone too much, they will feel overwhelmed or simply will start thinking that you are just trying to manipulate them emotionally to get something from them (that's exactly what happened on this video).