Tofu is a staple in many vegetarian dishes as well as in oriental cuisine. In essence, tofu is a "cheese" that is made from soy milk instead of cow milk. Soy is a type of bean. The word "tofu" in Japanese means "bean curd".
Soy and tofu are prized by vegetarians because it has good quantities of some types of protein in it, while not being a meat substance. It also contains iron.
A typical cube piece of tofu is about 2g in carbs. This means that having one or two tofu cubes in egg drop soup might only add a few carbs to your meal - but having an entire plate of 20 or more cubes can rack up the carb count quite quickly.
It's important to note that soy products have been linked to increased rates of cancer that involve estrogen. There's a compound in the soy that mimics the estrogen in humans. I won't delve into that entire argument here - if you are making the decision to be vegetarian, you should discuss all of these issues in depth with your doctor to determine the best way to keep your protein intake adequate. It is absolutely necessary for your long term health. You and your doctor can decide what is most appropriate when it comes to tofu and soy.
So, that being said, adding a few cubes of tofu to your egg drop soup is going to cause little problem for your low carb diet.