Exhale.... Inhale... Exhale...
Okay here we go. This post is only meant for pre-meds. Once someone has started down the path in medical school and beyond, I will only comment on specific cases, not generalities.
Your objective in life should be to be as happy as possible for as much of your life as you can. Medicine should be a rewarding career. Money is important, but your happiness is more important. I have long maintained that medical training is too long to be worth it if a) you aren't going to enjoy what you do at the end and b) at least most of the time enjoy the journey from undergrad to medical school to residency to fully licensed physician. Yes, it is the most surefire way to making it into the top 3-4% of earners in the United States, but there IS opportunity cost, both financial as well time wise. It is also important to keep in mind that while there is tremendous opportunity out there to make a lot of money in medicine, the 'easier' avenues are going to be shut down slowly but surely as our costs of healthcare continue to rise and people look at ways to save on healthcare. Simply put, if you don't derive pleasure from the day to day in medicine, you should not go into it.
With all of that in mind, patient's illness is always going to be a part of the job. The old adage, 'no matter how bad of a day you are having, their day is worse, after all they are sick and in the hospital, while you are at work' is not far from the truth. Obviously specialty and practice dependent, but you are interfacing with people who are dealing with the big things in their lives, every single day. Just as different specialties and practices will vary, so will specific patients. In general, the hardest are the younger patients who are going through major healthcare issues. Pediatrics is an obvious source of this, but something as different as vascular surgery will have similar cases. It is always hard to see 20 something and 30 somethings with kidney failure, on ventilators, or in shock. The other challenging group are the unexpected patients. Trauma falls into this, or the very health conscious patient that gets something they couldn't prevent. If these patients don't affect you at all, you are abnormal. I will stop short of saying that you 'shouldn't go into medicine'. But, I would strongly urge you to think long and hard about the road ahead. The flip side of that empathy is probably the most rewarding aspect of medicine for most physicians. The joy of being able to help people with their worst problems is one of the best highs one can attain.
This kind of emotional investment in patients can be dangerous as others in this thread have alluded to. Learning and establishing good coping mechanisms is mandatory. Understanding your overall ability to handle the highs and lows is equally important and should play a role into picking the specialty and practice that you ultimately end up in.