There is no simple way to answer this question, as it differs from case to case. As so much is dependent on the environment that the teeth are located in. If a tooth that under different circumstances could still be saved is surrounded by periodontal tissue that is infected or in poor condition, an extraction may be the best way to go. There are some simple things that are easy to spot, and I will talk about them, however, be warned that this blurb is no substitute for the opinion and diagnosis of a trained medical professional, and you should always get a consultation form an implantologist before going ahead and planning any sort of dental implant treatment.
If a tooth is so deteriorated that it can no longer hold a crown, even with a post and core made for it, it is time for a dental implant. If the tooth is dead, and the nerve is no longer living, than it may be unable to to be root canalled or it may be so structurally compromised that it can no longer hold a crown, it is time for a dental implant. If the tooth is so broken or damaged that it is no longer visible above the gumline, or there are possible fractures going deeper, beneath the gumline, it may be time for a dental implant.
On the whole, dentists will do everyíthing in their power to save a living tooth and not have to extract. That is because once you have missing teeth, your oral hygiene gets very complicated. To learn more about tooth loss, see the appropriate part of this website.
All in all you need to speak to an implantologist in order to find out if you need a dental implant to help you with your problem, or if your tooth can be saved.