Here are a few tricks that I've accumulated over the years to help improve relationships and convince others. I've tried most of them personally, and I've read up on the ones that I haven't. Message me if there are any good ones I've missed, or if these have worked for you as well!
Most people hate awkward silences during a conversation, but you can use them to your advantage. For example, when trying to make a sale, stop talking after making the initial pitch (e.g., listing prices). The more you talk, the more opportunities you give someone to say no. Additionally, when trying to get information out of others, stay silent if they give you just a partial answer. They're likely to continue talking to fill the silence, and you're likely to get the information you want.
This sounds obvious, which is why it's often overlooked; but smiling, especially while maintaining eye contact, really helps create a good first impression. When you see acquaintances, turn it up a notch and greet them with enthusiasm. Think of your friends, and of people who are always happy to see you – don't you end up feeling happy to see them as well? This is something that I should do more often myself - after all, the best way to get people to be comfortable around you is to show them that you're genuinely happy to have them around.
Ever wondered why the smile-and-nod always seems to work? Studies have shown that when people nod while listening to something, they're more likely to be in agreement. That makes sense- but how does it help you? Well, coupled with the fact that we tend to mirror someone else's body language, nodding while you're talking enables your listener to pick up on those physical cues. As they unconsciously nod along with you, they're more likely to end up convinced.
A friend once told me, "don't ever get mad at inanimate objects," which is sage wisdom, since we often forget that doing so just leaves us worse for wear. The same goes for interactions with people: When others are angry at you, the best tactic is to remain calm. Not only do they lose ground when they fail to provoke a response, but you come away as the reasonable party. Remaining calm also helps prevent a fight-or-flight response, leading to more logical decision-making on your part.
Know when to ask for favors.
This is a pretty famous one, originally attributed to Ben Franklin. As the story goes, Ben Franklin asked a rival to do him a favor (to lend him a rare book, in the original case), so as to improve their relationship. The reasoning is that once someone has done you a favor, they come to believe that they like you more- because why would they do a favor for a person that they hate? Cognitive dissonance in action. I haven't tested it yet, so if you don't like me, shoot me an email! I'd love to give it a try.
Go big or go home.
Speaking of asking for favors, it also helps to prime your target by first asking for something absurd. For example: if you need to borrow $50, first ask for $500. Not only does it create an anchoring effect, it reduces the chances of your being turned down the next time. People are less likely to turn you down twice, especially when the second request is much smaller than the first.
Keep some gum handy.
I don't chew gum enough to have tried this one, but apparently chewing gum when nervous helps calm you down. In fact, eating anything when nervous helps; chewing gum just happens to be the subtlest option with the fewest calories (provided that it's sugar-free gum, of course). Our brains have evolved to associate eating with a lack of danger, so chewing gum tricks us into thinking that we have nothing to worry about!
If you don't have any gum, take a 5-minute bathroom break to practice your power poses. Research has shown that striking poses associated with confidence and power actually makes you feel more confident and powerful. It might feel a bit silly at first, but I've found that it really does help. Possible stances include the "Superman" (hands on hips, chest out) and the "YMCA" (arms out, legs together, "Y" shape). Bonus application: when your alarm goes off, jump out of bed and do a power pose! It (usually) helps me avoid crawling back into bed.
Most mirroring happens unconsciously, when we're agreeing with or are attracted to someone. However, as mentioned earlier, it's possible to use it intentionally. By echoing other people's behaviors, mannerisms, and body language while you're with them, you're more likely to gain their favor. Underplay this one, since copying them too much would just look creepy; pick up a couple of hand gestures or expressions and see how that changes the flow of conversation. I assume this effect stems from a feeling of validation (and a corresponding boost in self-esteem) that comes as you follow their lead.
Teach to learn.
I've been using this one for years, although I'm definitely not the first to have thought of it. If you really, truly want to solidify your knowledge of a topic, teach it to someone else. It forces your brain to consolidate its information on the subject, and makes it much easier to recall.
Be a (modified) echo.
Repeat back what other people say to you-- not verbatim, of course, but as a summary of their main points (possibly as a question). By engaging in reflective listening, other people feel that they have been truly heard, and are more comfortable talking to you. This is a staple technique for a lot of management and relationship books because it improves relationships and makes others more likely to listen to what you have to say. On top of that, a bit of reflective listening goes a long way in avoiding miscommunications.
 Bonus points if you got this reference.
 Protip: If you're on a date, and he/she is mirroring your position/body language, that is a Good Thing.