Duty to rescue is a concept that many law systems have: In certain cirumstances, someone can be held responsible for not helping another person who is in danger, and who would be injured, or who would die, if not helped. People are required to help to the best of their abilities. As a bare minimum, this often means alerting emergency services. 

Not all law systems have this concept, and in general the following is true:

 The person helping must be able to recognize the situation
The danger must be immediate: That is: not helping will mean injury or death
 There is no requirement to help, if heping would put the helpers in danger.
 The duty may also exist in the case of "special relationships" (such as: being part of the same family, being married, working at the same company...)
Some professions (such as healthcare professional, midwives,...) have an extended duty to help

Those helping are often protected by law, against errors they make.

In many cases, the duty to rescue has become customary law. This is the case for example for helping people at sea.

Moral duty to help
A moral or ethical duty to rescue may exist even where there is no legal duty to rescue.  Such a duty can be justified in different ways.

One kind of justification is general and applies regardless of role-related relationships: Under this general justification, people have a duty to rescue other people in distress because of their common humanity. With this justification, the specific skills of the rescuer or the nature of the victim's distress are irrelevant. 

This kind would justify cases of rescue and in fact make such rescue a duty even between strangers.  With this justification in mind,  philosopher Peter Singer suggests that if one saw a child drowning and could intervene to save him, they should do so, if the cost is moderate to themselves. Damage to their clothing or shoes or how late it might make them for a meeting would be an insufficient excuse to avoid assistance.  Singer goes on to say that one should also try to rescue distant strangers, not just nearby children, because globalization has made it possible to do so.  Such general arguments for a duty to rescue also explain why after the 2010 Haiti earthquake, Haitians were digging family members, friends, and strangers out of the rubble with their bare hands and carrying injured persons to whatever medical care was available.  It also explains why, journalist and physician Sanjay Gupta and a number of other MD-journalists began acting as physicians to treat injuries rather than remaining uninvolved in their journalistic roles. Similarly, they justify journalist Anderson Cooper's attempt to shepherd an injured young boy away from some "toughs" nearby in the aftermath of the Haiti earthquake.

Specific arguments for such a duty to rescue include:

 The Golden Rule: treat others as one would wish to be treated.  This assumes that all persons would wish to be rescued if they were in distress, and so they should in turn rescue those in distress to the best of their abilities. What counts as distress requiring rescue may, of course, differ from person to person, but being trapped or at risk of drowning are emergency situations which this position assumes all humans would wish to be rescued from.
 Utilitarianism: utilitarianism posits that those actions are right which best maximize happiness and reduce suffering ("maximize the good").  Utilitarian reasoning generally supports acts of rescue which contribute to overall happiness and reduced suffering.  Rule utilitarianism would look not just at whether individual acts of rescue maximize the good, but whether certain types of acts do so. It then becomes one's duty to perform those types of actions. Generally, having strangers rescue those in distress maximizes good so long as the rescue attempt does not make things worse, so one has a duty to rescue to the best of her or his ability as long as doing so will not make things worse.
 Humanity: the rules of humanity advise that the essence of morality and right behavior is tending to human relationships. Therefore, virtues (desirable character traits) such as compassion, sympathy, honesty, and fidelity are to be admired and developed. Acting out of compassion and sympathy will often require rescue where someone is in need. Indeed, it would not be compassionate to ignore someone's need, though the way one fulfills that need may vary. In cases of emergency, rescue would be the most compassionate act compared with allowing a person to remain trapped in rubble.

There are also ethical justifications for role-specific or skill-specific duties of rescue such as those described above under the discussion of U.S. Common Law. Generally, these justifications are rooted in the idea that the best rescues, the most effective rescues, are done by those with special skills. Such persons, when available to rescue, are thus even more required to do so ethically than regular persons who might simply make things worse (for a utilitarian, rescue by a skilled professional in a relevant field would maximize the good even better than rescue by a regular stranger). This particular ethical argument makes sense when considering the ability firefighters to get both themselves and victims safely out of a burning building, or of health care personnel such as physicians, nurses, physician's assistants, and EMTs to provide medical rescue.

References 

Law