The term starburst conveys the dual notions of intensity and limited duration. There is no strict definition, however, so the term has been used (or abused) to encompass a huge variety of star formation events. Several speakers proposed useful definitions, and I will follow their lead.
Significant star formation is a hallmark of about half the galaxies in the local universe. There are several convenient proxies for active star formation: blue optical/UV colors, emission lines, or strong infrared output. Although there was a general awareness of the statistics, the Sloan Digital Sky Survey has recently brought home the unmistakable bimodality of optical colors: galaxies fall into either a red or blue sequence (separated by about 0.4 mag in B-V color), with few systems in between. This means they are either active star formers (blue) or have not hosted significant star formation for 1 Gyr. Interphase types are rare because once star formation ceases, color evolution from the blue to the red sequences occurs in only ~ 500 Myr (as long as the active galaxy is itself old).
The blue systems are mainly disk-dominated. The normal structure and dynamics of disks favor relatively slow conversion of gas to stars. Global self-regulation within disks is evidently effective over long timescales because there are good correlations between ionizing populations (lifetimes 10 Myr) and broadband optical colors (characteristic times of 1000 Myr). The range of what might be called "quiescent" star formation encountered along the normal Hubble sequence is about 4 orders of magnitude in both star formation rate (, measured in M yr-1) and star formation surface density (SFR, measured in M yr-1 kpc-2). A significant number of local galaxies, mostly dwarfs, exhibit elevated activity, ranging above SFR ~ 0.1, which might be called "enthusiastic" star formation. This accounts for only ~ 15% of all local star formation.
The most interesting cases, naturally, are the "psychopathic" ones at the extremes, which are the starbursts. These are often, though not always, associated with a large disturbance to normal disk kinematics. The central feature of a starburst is the concentration of star-forming activity and especially the large feedback it produces on its surroundings, often driving a "superwind" out of the host galaxy. For definiteness, I will define a "major starburst" as an episode where such effects are important. This requires SFR > 1 M yr-1 kpc-2, equivalent to L(BOL) > 1010 L kpc-2 or about 1000 × normal values for disk galaxies. The relevant quantities must be averaged over a finite cell size in area and time (say 1 kpc2, 10 Myr), and the initial mass function (IMF) must contain massive stars capable of producing ionization, winds, and core-collapse supernovae.
The fact that this definition is arbitrary is actually significant: the properties of star formation regions appear to be continuous across the range of amplitudes observed.
Starbursts constitute an important fraction of all detected high-redshift galaxies. Because of the fierce dimming effects in distance modulus and surface brightness at large z, there is a powerful selection effect operating here. Nonetheless, statistics based on co-moving volume densities have shown that while major starbursts are rare locally (~ 0.5% of nearby systems) they were much more common at earlier times (perhaps 15% in number and 70× in luminosity density at z = 1). IR-bright starbursts have been responsible for as much as 80% of the local stellar mass.
Although we don't have a definitive understanding of starbursts and their effects yet, the wealth of new data is making progress rapid. Along the way, there are two central difficulties: (1) major starbursts are rare in the local universe, and we are forced to scale up our understanding of physical processes from local samples and conditions; and (2) starbursts are notoriously complex 3-D systems, made especially difficult to probe by often severe differential internal extinction.