The scariest part of a nail trim is the possibility of hitting the "quick" - the blood vessel that is located in the growing nail. Just like when you trim your nails too short, if this part of the nail is cut there is a small amount of pain...and some blood. Although it can seem like a LOT of blood, there is no need to panic as the bleeding can easily be stopped using an over the counter styptic powder (like Quik-Stop) or applying a small amount of baking powder or cooking flour. White or light colored nails are easier to trim as the quick can be seen while black toenails require a little more caution to avoid hitting the sensitive area.
Most small dogs can be restrained by holding them like a baby and large dogs generally do best lying on their side while getting a belly rub. Use a towel to make a "kitty burrito" to prevent bites and scratches from a fractious cat. Gently hold each paw while taking a small amount of nail off at a time to prevent cracking the nail and cutting too high. The goal is to take the sharp point off the nail or to reach the "target" inside the nail.