Trimming a dog's nails can be dangerous for both dog and human if the dog is afraid. Dogs who fear nail trims can thrash around wildly, increasing the risk of cutting the quick if they are not effectively restrained. Cutting into the quick (the sensitive tissue within the nail) will only exacerbate the dog's fear. Additionally, dogs that are very afraid or in pain are likely to bite, presenting a safety risk to the humans involved.
Nail trim aversion can result just as easily from improper socialization to husbandry/handling as from "quicking" a dog. Whenever possible, it is best to socialize puppies to all types of handling extensively from 3 to 12 weeks. It is always easier to prevent nail trim aversion (or any fearful behavior) by creating lots of positive experiences during this critical socialization period than to rehabilitate a dog with a well-established history of nail trim aversion.
Regardless of whether you have a young puppy with no nail trimming experience or an adult dog with an existing fear, the answer to the question of "how to trim your dog's nails" is to train your dog to love having her paws and nails handled.
Puppies, being socialization sponges at this age, will likely proceed through the training very quickly. Dogs with established reactivity to the procedure may require more time. Be patient with your dog, and use really yummy treats (cheese, liver, hot dogs, meatballs, or liverwurst, whatever she LOVES).
How To Trim A Dog's Nails
Once you have taught your dog to love having her paws and nails handled, it is time to get clipping! If you have never clipped a dog's nails before, it may be helpful to ask your veterinarian or groomer to demonstrate the correct way to do it. Black nails are a bit trickier than white/clear nails to clip, as you cannot see the quick.
It's best to make sure your nail clippers are sharp. Dull, old nail clipping blades should be replaced before nail trimming as they are more prone to crush the nail than give a nice, clean clip.
Here is a great article from the College of Veterinary Medicine at Washington State University on nail clipping. It describes the two most common types of clippers (guillotine style and scissors style), along with tips on how to hold and use each of these tools. Using lots of photos, the entire process of a nail trim is described. If you have done the training laid out in Laura's videos, you will likely not need to restrain the dog as is mentioned in the WSU article.
Non-Clipper Techniques For Nail Trimming
Dogs that live in urban environments and are walked frequently on pavement may require nail trims very rarely, as the pavement acts as a nail file during walking. However, these dogs will likely require trimming of the dewclaws, as these nails are higher up on the foot and do not make contact with the asphalt. The article from WSU gives great advice on using the scissors style clipper to trim lengthy dewclaws.
Also popular are Dremel-type tools for nail clipping. Dremels should be introduced slowly (as should the clippers), and Laura's video above will work well for getting a dog to like having her paws handled regardless of the tool you use. Here is a great article on how to get your dog used to the nail clipper and eventually, use the Dremel to trim nails.
Finally, clicker-savvy dogs can be shaped to use a "scratchy board." You'll need a sturdy board, some sandpaper, and a staple gun to create your scratchy board. Staple the sandpaper so that it surrounds the boards, and shape for paw contact and eventually, scratching against the board. (Here is a video demonstrating a dog trimming his own nails with a scratchy board.) Whichever method you choose, with a little time and effort your canine can be well on her way to calm, safe nail trims. Good luck and happy trimming!