The answer to whether you should apply for a trademark or a patent depends on what you're trying to protect.
A trademark protects a brand's image and reputation against infringement. A trademark usually protects the company/product logo, name or slogan and prevents others from using brand elements that could be considered "confusingly similar", which could lead to consumer confusion.
A patent is intended for the protection of the technical aspects of an invention.
Bluetooth is an example of a patent (or a series of patents, to be exact). But it is not the name Bluetooth that the patent protects - the actual title of the main patent no.6590928 is "Frequency hopping piconets in an uncoordinated wireless multi-user system". The name "Bluetooth" is protected by a trademark.
You can learn more about the differences between common intellectual property protection mechanisms in this article.