Most places want 3 at least. With 3 letter, it may be ideal to get Two from neurologists, perhaps also your schools neurology head. The third could be any physician who would recommend you as a strong applicant. Although, some believe that maybe an internal med letter may be best since you will need to secure a prelim year in internal medicine to complete before starting neurology. So, with a good rec from an internist, that seems logical, but certainly not a must.
A good question. Of all the parts of your application, your letters best compliment your interview and can really effect how you are assessed after they have decided to interview you.
This isn't always crystal clear to predict, but anyone that knows your work well that can give you a stunningly positive letter is a big asset. In my limited experience, a key is to have the letter be sparkling. Comments like "the best student I have ever worked with" or "unparalleled potential in x, y and z" helps a lot more than a form letter with your name thrown in.
It's even better if that person is well known in academics (if you are applying to academic programs). There are relatively few super well known physician scientists or clinical leaders at a national level. So everyone ends up knowing each other and a lot of people are old friends or trained with or under each other. An exceptionally strong letter by someone every knows gets brought up in most interviews and can really help you stick out in the mind of the interviewer before and after the interview.
One trick is to ask around your school and see who has a rep for writing great letters. Graduating students start to get a good feel for this because it's not uncommon for interviewers to read part of your letters to you when you interview. You also need to politely ask each of your letter writers if they feel comfortable writing you an outstanding letter of recommendation.
To summarize: #1 All your writers should be willing to write you a great letter. #2 At least one neurology staff should probably be someone that worked closely with you clinically. #3 Try to get a letter from a senior faculty in neurology or closely related field that thinks your outstanding. #4 Research or clinical faculty that hold you in high regard are also valuable.