you can't pull half of a tooth. it has to come out root and all. depending on how bad your teeth are they may be able to save the structure with a root canal and then crown. if they are working on several teeth at once i know many dentist offer the option of local anesthesia (just numb the gums) or general anesthesia (put you to sleep). you'll have to make an appointment with a dentist and discuss your options.
If you feel that its necessary, you may want to consult with your dentist about finding an oral surgeon who can perform the extractions under sedation. If you have need for extensive extractions, it may be more comfortable that way. Also, if you are overly anxious, you may need to be sedated.
The teeth that need to be extracted vs the ones that can be saved by a root canal and crown will need to be determined by the dentist.
Like Edward84 said, you can't leave the half of the tooth in the jaw bone as it can be prone to infections.