The limbic system, named after the Latin word limbus for edge, is the innermost part of the brain, wrapped around the core ventricles. It is filled with cerebrospinal fluid and various clumps of white matter, which does not play much of a role in cognition.
This system is called the "old mammalian system" or the "mammalian brain," in the popular triune brain model, which splits the brain into three parts depending on their location and functions. The other parts are the reptilian brain or the brain stem, and the cerebral cortex or the neocortex. These are responsible for "lower" and "higher" behavior respectively.
The limbic system's components are the amygdala, the hippocampus, the cingulate gyrus, fornicate gyrus, hypothalamus, mammillary body, epithalamus, nucleus accumbens (the brain's famed "pleasure center"), orbitofrontal cortex, parahippocampal gyrus, and the thalamus. Each plays an important role in making things run smoothly in the brain. Analogous structures can be found in almost all mammals such as dogs, cats, and mice, though not in reptiles, which only possess a brain stem.
The limbic system is the home of emotions, motivation, the regulation of memories, the interface between emotional states and memories of physical stimuli, physiological autonomic regulators, hormones, "fight or flight" responses, sexual arousal, circadian rhythms, and some decision systems. It is what gets "duped" when people get addicted to hard drugs. Because the addiction happens in the "lower," "preconscious" portion of the brain, we cannot rationally consider its effects, and therefore recovery and relapse avoidance can be difficult. Rats given switches connected to electrodes which electrically stimulate their nucleus accumbens will continue pressing the switch at the exclusion of all else, including food or sex.
On top of the limbic system is the cerebral cortex, the "thinking brain." The thalamus acts as a liaison between the two. The cortex evolved dependent on the limbic system, which was present before it. Every beneficial adaptation in the neocortex had to "play nice" and interoperate efficiently to justify its own retention through improving the overall fitness of the organism. The pineal gland, a famous part of the limbic system located in the epithalamus, is a rare example of a vestigal brain organ, which was much larger and differentiated in an earlier part of our evolutionary history.