When to Start:  It's best to start trimming a dog's nails as a young puppy, that way they will get used to the process and be used to you touching their feet.  Sometimes, however (like with rescues), we don't have to opportunity to trim a dog's nails when they are young, and they may come to us with a predisposed hatred for having their nails trimmed.

Desensitizing the Feet:  You may need to start when your dog is asleep.  Simply rest your hand on their paw.  If they wake up, remove your hand and pretend you didn't notice.  Eventually you should be able to rest your hand on their paw without them waking up.  In really stubborn cases, you may have to start further up on the leg and work your way down toward the paw.  Once they'll allow you to keep your hand on the paw, start fiddling with their toes, moving them apart or squeezing the toenail.  Also get them used to you touching their feet while they're awake.  Once the dog's feet aren't sensitive to your touch anymore, it may be time to start with the clippers.

Desensitizing the Dog to the Clippers:  Now that the dog will allow you to touch their feet, bring out the clippers.  Let them get used to the sight of the clippers, and give them a treat if they sit quietly.  Work your way into clicking the clippers and making the sound.  Praise them and give treats.  Touch their toenail while clicking the clippers (don't clip the nail yet), and give a treat.  This may take several days or even weeks. 

You may choose to use a dremel sanding tool.  The process is the same.  Let the dog get used to seeing the tool in your hand, and let them get the idea that "dremel means treats".  Then turn the tool on and let them get used to the sound of the tool.  Eventually hold the tool in one hand and the dog's paw in the other.

Clip One Nail:  Now, use the clippers or dremel tool.  Give the dog a treat for seeing the tool (a good chewy treat is best, one that might distract them for a minute), and then clip one nail at the end (nowhere near the quick) or just touch the dremel to the nail.  Then praise and give another treat.  This may be the end of this session.

Clip Multiple Nails:  After the dog will let you clip one nail, work your way into clipping that nail closer to the quick (don't cut the blood vessel, though).  That may be it for today; if not, clip a second nail.  After a few sessions you should be able to do a whole foot at a time, with a treat after each cut. 

Don't go too far too fast.  If the dog will only let you do one nail or one foot at a time; that's fine.  Come back tomorrow and try again.  Be patient, but be firm.  And it works best if you use the best treats during nail trimming, treats they don't get at other times.  Work your way into being able to trim all four feet in one session.

Receding the Quick:  Sometimes the nails have grown so long that even trimming near the quick leaves nails that touch the ground.  After nails are trimmed, the quick will recede a bit.  Wait a few days after the first trimming and trim the same nail again.  Eventually it will get short enough that it doesn't touch the ground.