Trimming your cat's nails doesn't need to be traumatic for either of you!
The younger you can start your cat out with regular manicures, the better - but if you have an older cat who is squeamish about it, there's hope for them too.
First, desensitize your cat with every aspect of the procedure.
Break down nail trims into tiny steps:
Pick up your cat and place him on your lap facing out
Crouch down a bit and hold your arms by your side to "cradle" him in your lap
Let him smell the nail trimmer
Gently touch the foot with the nail trimmer
Grasp the left leg and hold it out where you can see the foot
Gently press the nails out, one at a time
Trim one nail at a time - don't forget the dewclaw ("thumb" that grows on the inside of the leg
At any point if your cat gets very anxious, stop the process and start counter-conditioning and desensitization one step at a time. This means you will pair your cat's favorite treat (try deli turkey) with each step of the process. Pick your cat up and place him on your lap facing out - then treat. Repeat many times. Once your cat is comfortable with this step, move to the next one: pick your cat up and place him on your lap facing out, then crouch down and cradle him in your lap - then give a treat. Repeat many times and once he's comfortable with step one and two in succession, move to the next step.
It is good to leave grooming tools out so your cat gets used to seeing and smelling them. I recommend using these nail trimmers so you don't have to worry about over-trimming. They have a very sharp conical blade that cleanly cuts the nail, rather than crushing it. You simply get the nail into the hole and squeeze. They come in different sizes - XS for a young or smallish cat, SM for a larger cat, MED for an older cat with very thick nails.
I did a video for you so you can have a visual of the steps above: