We demonstrate that the radiation emitted by ultrarelativistic electrons in highly nonuniform, small-scale magnetic fields is different from synchrotron radiation if the electron's transverse deflections in these fields are much smaller than the beaming angle. A quantitative analytical theory of this radiation, which we refer to as jitter radiation, is developed. It is shown that the emergent spectrum is determined by statistical properties of the magnetic field. The jitter radiation theory is then applied to internal shocks of gamma-ray bursts (GRBs). The model of a magnetic field in GRBs proposed by Medvedev & Loeb in 1999 is used. The spectral power distribution of radiation produced by the power-law-distribute electrons with a low-energy cutoff is well described by a sharply broken power law: P(omega) proportional to omega(1) for omega less than or similar to omega(jm) and P(omega) proportional to omega(-(p-1)/2) for omega greater than or similar to omega(jm), where p is the electron power-law index and omega(jm) is the jitter break frequency, which is independent of the field strength but depends on the electron density in the ejecta, omega(jm) proportional to n(1/2), as well as on the shock energetics and kinematics. The total emitted power of jitter radiation is; however, equal to that of synchrotron radiation. Since large-scale fields may also be present in the ejecta, we construct a two-component, jitter + synchrotron spectral model of the prompt gamma-ray emission. Quite surprisingly, this model seems to be readily capable of explaining several properties of time-resolved spectra of some GRBs, such as (1) the violation of the constraint on the low-energy spectral index called the synchrotron "line of death," (2) the sharp spectral break at the peak frequency, inconsistent with the broad synchrotron bump, (3) the evidence for two spectral subcomponents, and (4) possible existence of emission features called "GRB lines." We believe these facts strongly support both the existence of small-scale magnetic fields and the proposed radiation mechanism from GRB shocks. As an example, we use the composite model to analyze GRB 910503, which has two spectral peaks. At last, we emphasize that accurate GRB spectra may allow precise determination of fireball properties as early as several minutes after the explosion.
