Nursing is the science and art of providing care to the sick, injured, and infirm. A nurse is someone trained and licensed¹ in that practice.
The US recognizes two kinds of nurse. Their exact responsibilities and scope of practice vary, as specified by each individual state's Nurse Practice Act, but the basics are approximately the same across the country.
A Licensed Practical Nurse (LPN) typically has one year of career training before licensure. LPN education focuses on how to care for patients in stable condition with predictable outcomes. They can do the great majority of physical nursing tasks, such as passing medication, dressing a wound, placing a foley catheter, and so on. However, they receive little background in the underlying sciences of disease and medicine. For that reason they are not permitted to perform tasks that require assessment and independent action, such as administering blood products or pronouncing death. An LPN works under the direction of physicians and RNs.²
A Registered Nurse (RN) earns a college degree before licensure. In addition to performing all the same physical tasks as an LPN, they have more independence, and greater leadership and decision-making responsibilities. An RN does assessments, makes nursing diagnoses, designs the care plan, provides education and counseling, and delegates tasks to LPNs and unlicensed personnel. They provide care to all patients, including those whose condition is unstable or unpredictable. RNs are considered professionals in their own right; though they work under doctors in providing medical care, they act independently in providing nursing care.
The RN may have either an Associate Degree in Nursing (ADN) or Bachelor's of Science in Nursing (BSN). The BSN provides a stronger academic background, especially in sciences such as pathophysiology, pharmacology, nursing theory, research, and so forth. However, as both degrees lead to the same license, they legally provide the exact same responsibilities and capabilities. The BSN-prepared RN is often given preference in hiring or promotion decisions, and may or may not be paid a higher rate than an ADN counterpart who does nominally the same job; there is an ongoing, heated debate about whether various parts of this situation are entirely fair.
Nurses may also obtain graduate degrees, and at that point there is a fork in the road. One choice is to obtain a nonclinical Masters of Science in Nursing (MSN) or a Ph.D. in Nursing, preparing for positions in administration, professorship, or academic research. The other choice is a clinical MSN or a Doctorate of Nursing Practice (DNP), leading to a higher level of licensure as an advanced practice nurse. (Nurse practitioners are an example of nurses who have followed this latter path.)
The most familiar image of the nurse is one who works in a hospital, directly at the patient's bedside, but this is far from the only option, and career options are enormously varied. Nurses may work in doctors offices, home health care, hospice, psychiatric care, infection control, immunization clinics, school nursing, public health, insurance companies, advice lines, telephone triage, disaster response, STI clinics, nursing informatics, or dozens of other specialties.
¹In most countries and all US states, the term "nurse" is legally protected. Only a person possessing an active nursing license may call themselves a nurse. Penalties for impersonation can be as high as several months in prison.
²Common sense does come into play here. Though an LPN with 20 years experience may find herself working "under the direction" of an RN fresh out of school, only a foolish RN would fail to take her lead from the more experienced nurse.