The reaction γ+p→N*(1238)+π- has been studied in a hydrogen bubble chamber exposed to a 4.8-6.0-BeV bremsstrahlung beam. This channel dominates the two-pion photoproduction reaction below the 1.05-BeV ρ0-meson threshold, and at higher energies accounts for a significant part of the pπ+π- events after subtraction of the pρ0 channel. Total and differential cross sections, and N*(1238)-decay angular correlations are presented over the energy range 0.5-1.8 BeV. These data are analyzed using a resonance model in which the reaction proceeds through the photoexcitation of the higher nucleon isobars-N**(1420), N**(1512), N**(1688), and possibly N**(1924). The only free parameters are the resonance strengths, which are fixed by a best fit to the total-cross-section curve. These same strengths account for the N*(1238) production angular distribution, but there remain some discrepancies between the predicted and observed N*-decay angular correlations in the lowest-energy regions covered by our fit. Thus, although the total cross sections and production angular distributions require an appreciable contribution from N**(1420) excitation, the decay angular correlations tend to minimize the contributions of this resonance. However, satisfactory agreement with all the data may be obtained with a downward shift of the N**(1512) (J=32-) resonance mass by ∼50 MeV. Comparison is also made between our data and the one-pion-exchange model with final-state interaction. Unless this model is drastically modified (as for example in the calculations of Stichel and Scholz), agreement with the observations is poor.