Last year we donated a 14-year-old Taurus station wagon to Goodwill. We found them through the auto-donation.com website, which allows you to choose among dozens of different charities. Goodwill arranged the tow truck (the Taurus was running but the tow truck was easier for them), and it left the house about a week later.
You sign up online and usually get an immediate yes/no answer on whether they want it. Before they pick up the car, you'll get donation paperwork either from the website or in the mail. You'll probably end up mailing away the title (in our case the title went to Maine for an Oahu donation) and just tossing the keys to the towtruck driver. I also felt obligated to hose the car off, pitchfork out the passenger compartment, and include a folder with the car's documents & maintenance records, but the charity might have specific requirements. The towtruck driver might give you a receipt for the pickup but they're just the contractor-- the charity's pickup receipt is probably your website printout.
Feels kinda strange to just drop a title in the mail. You spend over a decade caring for a piece of valuable paper, or at least something that's a hassle to replace. Then one day you have to decide that it's essentially worthless and you mail it off (along with your Social Security Number) to a bunch of strangers you met over the Internet.
You'll get a tax receipt a few weeks/months after you wave bye-bye to the tow truck. It depends on how often the auctions occur, so don't delay until November/December if you want the deduction in this tax year. The tax receipt will give you the gross proceeds (how much the car got at auction) and, depending on the auction price and the charity, it may even include a Form 8283. The charity also sends a 1098C to the IRS for them to computer-match to your return, so you report exactly what's on the tax receipt. If your itemized deductions are greater than your standard deduction then you'll probably be able to deduct the auction value. If not, well, then, I guess you did something nice for a charity.
We also considered the following charities but didn't research them:
Kokua in Kind
(Hawaii only, but there might be a similar charity in your area)