We propose an overview and unified analysis of Bregman proximal first-order algorithms for convex minimization.  Our approach highlights the fundamental but somewhat overlooked role that the Fechel conjugate plays in this important and versatile class of algorithms. Our approach yields novel proofs of the convergence rates of the Bregman proximal subgradient, Bregman proximal gradient, and a new accelerated Bregman proximal gradient algorithm.  We illustrate the effectiveness of Bregman proximal methods in two interesting applications, namely the D-optimal design and Poisson linear inverse problems.