The Human Resource Management (HRM) function includes a variety of activities, and key among them is responsibility for human resources -- for deciding what staffing needs you have and whether to use independent contractors or hire employees to fill these needs, recruiting and training the best employees, ensuring they are high performers, dealing with performance issues, and ensuring your personnel and management practices conform to various regulations. Activities also include managing your approach to employee benefits and compensation, employee records and personnel policies. Usually small businesses (for-profit or nonprofit) have to carry out these activities themselves because they can't yet afford part- or full-time help. However, they should always ensure that employees have -- and are aware of -- personnel policies which conform to current regulations. These policies are often in the form of employee manuals, which all employees have.
Some people distinguish a difference between HRM and Human Resource Development (HRD), a profession. Those people might include HRM in HRD, explaining that HRD includes the broader range of activities to develop personnel inside of organizations, e.g., career development, training, organization development, etc.
The HRM function and HRD profession have undergone tremendous change over the past 20-30 years. Many years ago, large organizations looked to the "Personnel Department," mostly to manage the paperwork around hiring and paying people. More recently, organizations consider the "HR Department" as playing a major role in staffing, training and helping to manage people so that people and the organization are performing at maximum capability in a highly fulfilling manner. There is a long-standing argument about where HR-related functions should be organized into large organizations, eg, "should HR be in the Organization Development department or the other way around?"
Recently, the phrase "talent management" is being used to refer the activities to attract, develop and retain employees. Some people and organizations use the phrase to refer especially to talented and/or high-potential employees. The phrase often is used interchangeably with HR -- although as the field of talent management matures, it's very likely there will be an increasing number of people who will strongly disagree about the interchange of these fields.
Many people use the phrase "Human Resource Management," "Human Resource Development" and "Human Resources" interchangeably, and abbreviate Human Resources as HR -- HR has become a conventional term to refer to all of these phrases.