The Erdős-Rényi process begins with an empty graph on n vertices, with edges added randomly one at a time to the graph. A classical result of Erdős and Rényi states that the Erdős-Rényi process undergoes a phase transition, which takes place when the number of edges reaches n/2 (we say at time 1) and a giant component emerges. Since this seminal work of Erdős and Rényi, various random graph models have been introduced and studied. In this paper we study the Bohman-Frieze process, a simple modification of the Erdős-Rényi process.The Bohman-Frieze process also begins with an empty graph on n vertices. At each step two random edges are presented, and if the first edge would join two isolated vertices, it is added to a graph; otherwise the second edge is added. We present several new results on the phase transition of the Bohman-Frieze process. We show that it has a qualitatively similar phase transition to the Erdős-Rényi process in terms of the size and structure of the components near the critical point. We prove that all components at time t c − ǫ (that is, when the number of edges are (t c − ǫ)n/2) are trees or unicyclic components and that the largest component is of size Ω(ǫ −2 log n). Further, at t c + ǫ, all components apart from the giant component are trees or unicyclic and the size of the second-largest component is Θ(ǫ −2 log n). Each of these results corresponds to an analogous well-known result for the Erdős-Rényi process. Our proof techniques include combinatorial arguments, the differential equation method for random processes, and the singularity analysis of the moment generating function for the susceptibility, which satisfies a quasi-linear partial differential equation.