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In an argument
First, you should listen to her side of the argument with respect and not just wait your turn. Even if you think she's wrong. Hear her out. NEVER TELL HER TO CALM DOWN. Any woman will tell you that will make her blood boil with rage. Ask sincere questions about the issue. Then, calmly tell her your side, starting with what you can agree on, then where you disagree, and then repeat what you said you agreed on. Consider where you can compromise. Repeat that process until the argument is resolved.
If you want to understand them
Men are very straightforward. Women talk from angles to avoid being impolite or rude or pushy. To understand women, listen between the lines. They are rarely blunt. For instance, if a man's wife wanted him to take her to the park she would say, 'It's been a long time since we went to a park', what she would mean is 'I want you to go with me to the park soon. ' But, ask first or feel her out with something like 'You're right. I'd love to go to the park with you. When should we go? '.
You want to win her over
If you want the relationship to last, be yourself. But be yourself while also keeping in mind that there are some ways women do and do not like to be talked to. Do: compliment her, ask her about her day or life, open up about your day or life or feelings, listen to her with your full attention. Don't: criticize her about small things (like - that's not how my Mom does it), pay attention to something else when she's talking to you, interrupt her in the middle of a story, or push her to do things.
To encourage her
If men want to help each other succeed at something, they will point out what's not working. DO NOT DO THIS TO A WOMAN. You can ask HER what SHE thinks isn't working and what SHE thinks will fix it. Don't offer suggestions. A lot of women will translate it as you thinking bad of them. Don't congralate her success with words. Take her to dinner or something to celebrate WITH her not for her or with gifts. Let her success be something she did for herself not for or to impress anyone else.
If you want to date her
A little teasing is okay when flirting but don't pick on her. Don't talk too much about yourself. Look for cues:like if she touches your arm when she's talking she might be into you. If she faces away, she's not. Get a clear answer. Don't ask her out for a specific day or event. If she says has plans already, she might be letting you down easy. Also, be clear it's a date. 'Would you like to go on a date next week? ' If she says no, you can move on. If she says yes, you know it's a real date.
Just socially or in business
Pretty much the same way you'd talk to anyone. Pay the same attention to what she says as you would to what a man says. Give her opinions the same level of respect. One thing, though - a lot of women will couch the way they say things. While a man might say 'Let's do XYZ on the project', a woman with the same idea would say 'What do you think of doing XYZ on the project? ' If you want to be popular socially or in business with women, listen to how they put things and then follow that pattern.