Dr is the usual abbreviation for 'Doctor'. This may be a medical professional who treats people in ill-health; it may be an academic. The word 'doctor' originally meant 'teacher'. Nowadays, the implication is 'one equipped to teach in Higher education', hence someone who has earned the highest degree offered in Universities. OED says that the title of 'doctor' is, "in popular current use, applied to any medical practitioner". Professionals in health tend to use 'doctor' for most practitioners; but some of the most prestigious medical staff, namely consultant surgeons, are known as 'Mr'. (Consultants who are physicians are known as 'Dr'.)
There are different rules of style regarding the punctuation of the abbreviation Dr. Traditionally British English taught that, where the last letter of a word survived in the abbreviation, there should be no full stop. An abbreviation should only end in a full stop when the last letter of the abbreviation was not the last letter of the full word. Some style guides, departmental Handbooks and so on say that Dr. should end in a full stop. Do what your reader wants!
In academic circles, doctor usually means the holder of a higher degree in any subject. A MusD, for example ('Doctor of Music') and an LlD ('Doctor of Law') are both named 'Doctor Smith'. (The order of the letters in the abbreviation is because the abbreviations are for the Latin form of the title: Musicae Doctor and Legum Doctor. The double 'Ll' is the conventional rendering of the plural, legum meaning 'laws'; the singular is lex.) The commonest two doctorates are the M.D. (Medicinae Doctor), the 'Doctor of Medicine'; and Ph.D (Philosophiae Doctor), or 'Doctor of Philosophy'. This last is not necessarily learned in the subject of Philosophy: like the 'Arts' in B.A., 'Philosophy' in PhD is a general term. It may be awarded to those with research skills in subjects like History, English or Chemistry. At Oxford University, and some others, the Latin title is given as D.Phil..