Have a position before you enter the room. Good negotiation always starts with knowing what you want and putting it forth in your initial conversation. The rest of the negotiation is convincing the other party that it's in their best interest to meet you there.
Make sure your position is realistic. If you wanted two million dollars and a platinum-plated Rolls Royce, negotiations would be difficult. Thankfully, if you're presenting an estimate for a design project and have some strong logic behind your pricing, or some ground as to why your audience will prefer green instead of blue for their new logo, there is less room for slippage. Speaking of which...
Have a fallback position that you can happily live with. And don't go there as soon as you need to give ground. Go only halfway.
Know your trade-offs. When you're negotiating around a project, it's rarely about just time and money. It can be about prestige, portfolio work, meaningful contribution to the world at large, and creative freedom. Know what you're bartering if your fees are cut for the promise of future work, and don't let it steal the bread right off your table. Stroked egos don't pay bills.
Always keep the big picture in mind. Changing a headline or swapping out a photo shouldn't be a drama. Spiking a killer concept to play it safe, that's another story. Know which decisions require formal negotiation, and which are just potholes in the road. If you have a strong creative brief and a strong contract, many of these points should be non-issues.
If the stakes are high, take your time. Keep in mind that the larger the negotiation, the more important it is to never show your "settling point" until you're out of your initial negotiation. Get some time and space to think it over. Otherwise, whomever you're negotiating with think that they can push you even further. (This is very hard for most people. They want the problem solved right there in the room, so they don't have to worry about it any more and can move on.)
Be prepared to leave the table at any point. Be willing to walk away from something you want if you think the terms are going to hurt you in the long term. This is very important in contract negotiations. Potential clients can smell desperation and will exploit it.
Keep your dialogue humane, respectful, and honest. When negotiating with clients or your boss, make sure you don't turn your negotiations into an "us vs. them" scenario. They share the same set of human needs that we do, and relating with them on a human level will not only strengthen your continued working relationship, but also cement the expectation that no matter what happens in your work, you'll always be on equal footing as human beings.
If you fail, don't take it personally. Just learn from it. Don't let your conscience eat a hole in your gut. Hindsight is 20/20. If you don't get the work because it wasn't the right fit, or the client overrules your beautiful color scheme because they don't like purple, then derive some learnings from what happened and move forward.
Well-couched failure can lead to a future win. Relationships between your co-workers and your clients continue, even if you fail in your negotiation. If you've been cordial and truthful about your position through the entire negotiation process, you'll gain respect, which is the coin that can barter you future success.