The Mobile Edge Computing (MEC)-based task offloading in the Internet of Vehicles (IoV) scenario, which transfers computational tasks to mobile edge nodes and fixed edge nodes with available computing resources, has attracted interest in recent years. The MEC-based task offloading can achieve low latency and low operational cost under the tasks delay constraints. However, most existing research generally focuses on how to divide and migrate these tasks to the other devices. This research ignores delay constraints and offloading node selection for different tasks. In this article, we design the MEC-enabled IoV architecture, in which all vehicles and MEC servers act as offloading nodes. Mobile offloading nodes (i.e., vehicles) and fixed offloading nodes (i.e., MEC servers) provide low latency offloading services cooperatively through roadside units. Then we propose the task offloading scheme that considers task classification and offloading nodes selection (TO-TCONS). Our goal is to minimize the total execution time of tasks. In TO-TCONS Scheme, we divide the task offloading into the same region offloading mode and cross-region offloading mode, which is based on the delay constraints of tasks and the travel time of the target vehicle. Moreover, we propose the mobile offloading nodes selection strategy to select offloading nodes for each task, which evaluates offloading candidates for each task based on computing resources and transmission rates. Simulation results demonstrate that TO-TCONS Scheme is indeed capable of reducing total latency of tasks execution under the delay constraints in MEC-enabled IoV.