There are two kinds of liability to deal with when selling your old car. You need to make sure that you convey any liability for future accidents to the new owner along with the vehicle title. Second, there may be a risk of liability if the buyer discovers car problems you didn't disclose.
Taking the buyer's cash and signing over the title does not, in many states, release you from liability if he gets into an accident or parks illegally in your vehicle. Your title form should include a release of liability you can submit to the state establishing that you're no longer responsible for the car. You should also contact your insurer and take the car off your policy, either by canceling it or transferring it to your new vehicle.
All states have "lemon laws" covering the sale of defective vehicles, but these apply more to dealerships than individuals. Check your state's law before you sell your car so that you know your obligations. In Massachusetts, for example, you have to inform buyers about "defects which impair the safety or substantially impair the use of the vehicle," in the state government's words. If the buyer can prove to the state that you knew about but didn't divulge a serious problem, you'll have to take the car back. The same applies if the car fails its state inspection.
When someone buys a car from a dealer, there's an implied warranty about the quality of the car. If someone buys from you, there's no implied warranty: The buyer takes the car "as is," regardless of its condition of defect, unless your state's laws explicitly say otherwise. This is one reason buyers expect a lower price from you than they'd get from a dealer: It's a tradeoff for the risk they're assuming.
If you sell or trade your car to a professional dealer, you'll probably have to sell at a lower price, but you don't have to worry about liability. As a professional, it's the dealer's responsibility to check out the car, and if the next owner has problems, it's the dealer who answers for them. Another alternative is to donate your car to charity and claim it as a deduction on your taxes.