There is no universal answer to this question, since everyone communicates differently. I can only share some tips that worked for me, and might work for other people.
Don't get known as "the vegan"
Avoid introducing yourself to other people, or in a conversation, as a vegan. If people are arguing about meat and how horrible tofu tastes, don't jump in as "Well, I'm vegan and I eat lots of tofu, and it's delicious."
I'm not saying "hide the fact you're a vegan". You shouldn't, since it's not offensive or something to be ashamed of or pitied for.
I'm saying "Don't reduce yourself to being a vegan". You're a person first, and a vegan second. When you talk to people, show them that you're an individual, and that veganism is not your defining property.
When I get to know new friends, my dietary choices are not the first thing they learn about me. They'll first get to know me, and later discover that I'm also a vegan. The same thing works on smaller timescales in a conversation.
So, be honest and authentic.
People don't want to talk to a sock puppet, they want to talk to a real person. Maybe I'm partially repeating the first point again, but it's important, so let me say it in different words.
Don't be militant or condescending
When I was a vegetarian, I knew a militant vegan who accused me of not being vegan yet. That's right, accused me. This caused a defiant reaction in me, and I didn't turn vegan until I forgot about him.
Most people don't want to be converted. Noone wants to adopt somebody else's viewpoints.
Corollary: People will start understanding your viewpoints if you give them space to come to your conclusions on their own.
Accept that there are other moral viewpoints
Some people just don't have (as much) empathy with nonhuman animals. And they won't understand your viewpoint. They're not wrong, they just have a different ethics which matches their empathy better. Sad, but true.
Try to understand these people, and make them understand, in turn, that their moral viewpoints are not unique either, and that you have a different viewpoint which isn't irrational either.
Don't always talk about veganism, and don't insist so much
Do you know that feeling when somebody is talking to you and you can literally feel the pressure in them to tell you about their opinions and views? Don't be that person. Don't give anyone the feeling that you need to talk about these topics.
Give people time to think, and space for their viewpoints
Don't frame the discussion such that everyone who isn't a vegan is automatically a Bad Person. Leave room for people to keep face!
Show how veganism is not a sacrifice
Talk about your positive experiences. And by that, I don't mean "talk about your experiences, and frame the in a positive way". No, don't talk about that time when you tried to make seitan and failed miserably as "a great experience". Be honest. Talk about what makes you a happy person. Talk about that time you made a really nice meal. And don't try to sell it like an advertisement. Talk about it because you would talk about anyways.
Don't enforce in-group-out-group thinking. Don't draw lines between vegans and non-vegans. Don't emphasize the differences. Don't point out what food isn't vegan, point out what food is vegan.
Lots of people say something like "I've never eaten a vegan meal before", "It's very hard to prepare a vegan meal" or "There is no vegan food at this place". Debunk these statements lightheartedly (while keeping the previous points in mind i.e. don't jump onto such statements with pressure). Show how a lot of food is just a minor alteration away from being vegan.
I hope some of that helped. If I'm not answering your question well, do drop a comment, and I'll expand.