Determinate and Indeterminate
Whether or not you prune off suckers is a personal preference, but it also depends upon what type of tomato you are growing. Determinate types are bushier and rarely require staking because they form flowers at the terminal ends of branches. This stops the plant from growing upward. Indeterminate plants are more lateral in habit and include the older varieties. The indeterminate type are more likely to form suckers that will cause the vines to get leggy and sparse and produce smaller fruit.
What is a Sucker?
A sucker is simply new growth at the "V" where a first leaf comes out of the parent branch. This could be off the main vine or another large branch. The sucker is small and spindly and is easy to pinch or snap off. Leaving the sucker on the plant will not harm the tomato and will produce a bushier plant. This is why some gardeners leave them so there is more foliage to produce plant sugars. The extra growth may crowd the flower buds and resulting fruit, which is why some gardeners remove them. It is up to you which way you cultivate your tomatoes.
Pruning or Pinching
You may use garden shears or pruners, or pinch out the secondary sucker growth by hand. Make certain whatever implement you use is clean to avoid spreading disease and pathogens. You do not need to remove every sucker, but on vines that have a good start it is recommended to allow them to fill out and grow thicker. Vines with little growth may require the suckers to stay on to fill in the foliage. Use your judgment on which and how many suckers to prune out. Cut the sucker where it attaches to the parent branch in the V. Do not cut into the parent or first growth branches.
Pruning is only one part of tomato care. Training is probably the most important aspect; use trellises, cages or any other vertical structure. Supply a large support structure with plenty of large holes to feed the vine into. Most determinate tomatoes get 3 to 4 feet tall, while indeterminate varieties can grow up to 6 feet. Add the weight of the fruit and you can see that a very sturdy structure is required. Yo can also use stakes with twine strung across in several layers.