Just to clarify, I do not at this time ask for any donations, nor have I ever done it before, so this text comes strictly from a user perspective. When you browse literally hundreds of websites daily, like I do, ads and donation requests that you actually notice must be doing something right.
1. Cuteness and good intentions
It may not seem like much at first, but Abandonia's donation link on the upper right side of the page seems like a perfect donation request to me. Abandonia is a site devoted to freely distributing abandonware games, so the cute pixel character hits the mark perfectly. It's subtly animated to drag your attention to it. It's practically begging for money, however it doesn't come of as cheesy because it fits into the site perfectly. Lastly, when you do click it, you see that the authors are spending all donated money only for web hosting costs, and if they raise more than they need, they give everything back to the users as prizes in various contests. I don't know how much donations they receive, but if this doesn't work, I don't know what does.
2. Be precise, make a progress bar.
When I work, I mostly listen to chillout/downtempo music for relaxation. One of my favorite web radio stations is www.lounge-radio.com, which is more or less completely donation-driven. Since they reached their donations goal for 2007 they changed the donation page to a thank you page, but I'm going to describe what it consisted of. It was a detailed explanation of how much money they exactly (down to the dollar) need, and what they will spend it for. They've made it perfectly clear that the site will cease to function if they don't get that sum until the end of 2006. They've made a progress bar, showing how much funds they've raised so far. And lastly, they included a list (which is still visible) of all donors. Needless to say, they easily got the required sum of money, which will enable them to keep on lounging for one more year.
3. No ads, only donations.
DropBoks, a simple free online file repository service, simplifies things to the extreme. They have no ads, the design of the site is very minimal. They make it clear through a small, unobtrusive yet visible button on the bottom of the page that the site is donation-driven, which enables them to have no ads whatsoever. So far, it works for them.
4. Be completely honest about everything.
Kottke.org, one of the most famous blogs on the net, currently has no ads and asks for no donations. However, when he needed donations, Jason Kottke wrote a very detailed post explaining exactly what he asks for and why he does it, going into somewhat personal details of his life and work. And it worked fantastic; the response was overwhelming.
5. Ask for something specific and important to your audience
This particular donation wasn't completely successful, but I still think it was a great way to attract attention and to communicate directly with your fans. Scott Kurtz, the guy who draws the comic at PVPonline, was constantly bombarded with fan-mail from Mac users who thought it would be cool if he bought a Mac. In this post, he asked them to put their money where their mouth is, asking for just one dollar from each visitor, which would be spent by him to buy various Apple hardware. He even included a cool progress bar showing how far had he got - as you can see, he barely made it to a 15'' Mac. Not perfect, but not bad also.