The asymptotic giant branch (AGB) is a grouping of stars (AGB stars) on the HRD that form a line close to the horizontal branch (HB) and often on a line similar to that of the red-giant branch (RGB). They are giant stars, commonly red giants.
AGB stars are in a phase of post-main-sequence stellar evolution in which the stellar core, already burning helium (which began in the HB phase), produces sufficient carbon and oxygen to settle into a central volume, "underneath" the helium. The result is a central region without enough helium to fuse, yet not hot enough to trigger carbon or oxygen fusion, i.e., the exact center has no fusion and the helium fusion (triple alpha process) continues (or resumes) only in a surrounding shell. (All this occurs in stars sufficiently massive to produce a core of helium and go through giant-star phases, the lowest mass allowing this being on the order a half solar mass during the star's main sequence.) Around this is a helium shell, surrounded by a shell of hydrogen burning. For very massive stars, the central temperature rises enough that carbon fusion begins, and depending upon the temperatures eventually reached (from sufficient mass), additional alpha-process (and similar) fusion steps eventually commence, forming additional shells of fusion around the center, the most massive stars eventually producing iron in the center.
During the AGB phase, the fusion may be unstable, i.e., depleting the fuel or losing the necessary temperature, later to gain these again, producing in multiple pulses of fusion (thermal pulses), an evolutionary stage termed TP-AGB for thermal pulse AGB (with the term E-AGB for early AGB indicating an AGB star before this phase). In some cases, the pulses are virtually hidden in the interior of the star: the process of transferring energy to the surface "smoothes over" the "bumps", in effect, buffering some energy, hiding the peaks in the production of energy from external visibility. In other cases, the pulses are visible, producing some of the many types of variable stars, e.g., Mira variables.
It isn't clear to me that writing is consistent regarding when the term AGB no longer applies to the star. Some use the term post AGB star, and one logical point at which to declare a star as post AGB is when it ceases growing more luminous, yet its surface grows hotter, thus no longer showing a grouping on the HRD similar to that of the red giant branch (as is suggested by the term asymptotic giant branch), and moving horizontally across the HRD. Beginning during the "proper" AGB phase, and throughout this subsequent evolution, the core is very luminous and radiation pressure causes a significant stellar wind (superwind), basically ejecting the giant's thin exterior volume, which becomes a circumstellar envelope (CSE). When fusion has ceased and the stellar core collapses further, its temperature rises, producing enough ionizing radiation to ionize the CSE until it glows (fluorescence), making it visible as a planetary nebula (PN).
For the most massive stars (above about 8 solar masses), the end result is core collapse, either a core collapse supernova, or if above about 40-50 solar masses, a collapse without the explosion. This final collapse produces a neutron star or stellar black hole.
For stars not this massive, when the PN has faded, the stellar core is left as a white dwarf.