#1: "Tell me more about it" 
Rather than jumping to conclusions about what you think people are trying to tell you and giving "unsolicited advice," encourage them to keep talking and opening up.
#2: "What are the reasons for your opinion?"
If you and the other person disagree on an issue ask this question before explaining your opinion.  This will make them feel at ease, rather than defensive.
#3: "I never thought of it that way"
It makes the other person feel as if they are influencing you.  Hello, ego boost.
#4: "That must have upset you"
It shows empathy and understanding while still remaining neutral.
#5: "How did it go?"
Follow up on things they tell you are coming up (i.e. job interview, wedding, etc).  It will show that you are thoughtful and a good listener.
#6: "You are a really generous person"
This is a genuine compliment that goes deeper than physical traits or surface qualities.  Always try to compliment someone's internal qualities -- it will mean so much more.
#7: "I really admire that"
Everyone wants to be admired, and be around people who admire them.   It's that simple.

Again, these were taken from a relationship article on Match.com, but I find that they are just as important in business and networking.  It's all about strong communication skills and understanding how people work.