The left-hand path, and right-hand path is a dichotomy based around two different ways of practicing occultism, esotericism, spirituality, and religion. Many definitions of these understandings that you will find are biased towards one of the paths, because of that here I will try to clear up the meaning of these words, and provide an objective definition for both sides that are not derogatory.
This understanding originated from Indian Tantra practices where the Tantra concept of vāmācāra became our understanding of the left hand path, and the Tantra concept of dakshinachara became our understanding of the right hand path. The Indian tantric understandings were translated, and brought over into the west by Helena Blavatsky in her book Isis unveiled which popularized its usage with occult practitioners. The traditional Indian Tantra understandings is still current in modern Indian, and Buddhist Tantra practices, though they are now very separate from our own understandings.
The left-hand path, and right-hand path understanding has been used by many different forms of practitioners, though some practitioners prefer not to use it, and may even see it as outdated. Some practitioners have even equated it to the black, and white magick paradigm, though it is distinctly different, and was only accidentally associated.
Both of these paths can be followed simultaneously, because they are not mutually exclusive though most practitioners are considered to fall into one of them over the other. Both of these paths are valid and authentic, and can be used in order to have an occult, esoteric, spiritual, or religious fulfilling practice.
Left hand path, or LHP:
In some understandings the left hand path is equated with the understanding of black magick, and is seen as malicious, though this is what it was accidentally understood as.
The left hand path is connected with breaking of taboo, and the abandoning of set morality. It also has been connected with self-deification, and antinomianism. Self-Deification being the elevation of the self by focusing upon the individual in strengthening them, especially to God-like statuses, and antinomianism being the rejection of authority, and social norms, especially religious authority, and dogmatic practices. From all of this the left hand path could be seen as a heterodoxical practice that strongly emphasizes more individualist approaches. The left hand path is also said to glorify more material forms of indulgence, and is focused more on personal power, and self-satisfaction through the practice of their path.
Right hand path, or RHP:
In some understandings the right hand path is equated with the understanding of white magick, and is seen as benevolent, though this is what it was accidentally understood as. The right hand path is connected with ethical codes that are greatly connected to the social norm of that time, and place. The understanding of the right hand path is also connected to that of the divine, and becoming close to that divine. The right hand path is more focused on spiritual attainment, and the ascending of being. They have a higher focus on deity, and religious authority, doctrine, and dogma, because of this the right hand path can be seen as an orthodoxical practice. Ideas surrounding the right hand path are those of community, ethics, judgement, divinity, and self discipline.