HR recruiting is concerned with getting competent new hires into the organization in positions where they will be effective and motivated. While they must make hiring decisions to meet current business demands, they also want to bring in people who will grow with the organization, thus the heavy emphasis many businesses place on college recruitment. A huge factor in recruitment is remaining in compliance with the many laws and regulations that affect the hiring process, especially those regarding Affirmative Action and other non-discrimination factors. Because recruitment results in the new hires, those laws and regulations that govern hiring are an important factor in where and how the organization recruits.
Training is handled very differently by organizations, from outsourcing (sending employees to outside training), to entire divisions or departments that design and provide training, or a mix of the two. Some training is a mandated part of employee performance (such as training in standards compliance) or has been negotiated by a union contract (employees cannot be fired for poor performance if they've not received formal training in a particular skill).
HR is concerned that the training meets legal obligations, is conducted appropriately, that employees are selected for training in a fair and non-discriminatory manner, and that employees are gaining job-relevant skills that keep the organization effective and productive.
Motivating employees to remain effective and productive involves knowing what actually motivates employees in general and making choices about how employee programs or decisions that affect employees will raise or lower their motivation. Should they implement raises or provide new furniture; provide a better phone system or a break room? Much has been written about employee motivation and studies abound. It's HR's job to review these and find out what its employees need to motivate them to perform. This is probably the most difficult aspect of the three factors.
All three factors require that HR actually understands the needs of the business, knows the department or division heads and how they operate, and listens to employees. These will help HR gain a better understanding of the business, the people who run it, and the people who do the work so HR can make better hiring and training decisions and create an environment that motivates people to work towards the organization's goals.