Terrestrial communication networks mainly focus on users in urban areas but have poor coverage performance in harsh environments, such as mountains, deserts, and oceans. Satellites can be exploited to extend the coverage of terrestrial fifth-generation (5G) networks. However, satellites are restricted by their high latency and relatively low data rate. Consequently, the integration of terrestrial and satellite components has been widely studied, to take advantage of both sides and enable seamless broadband coverage. Due to the significant differences between satellite communications (SatComs) and terrestrial communications (TerComs) in terms of channel fading, transmission delay, mobility, and coverage performance, the establishment of an efficient hybrid satellite-terrestrial network (HSTN) still faces many challenges. In general, it is difficult to decompose a HSTN into a sum of separate satellite and terrestrial links due to the complicated coupling relationships therein. To uncover the complete picture of HSTNs, we regard the HSTN as a combination of basic cooperative models that contain the main traits of satellite-terrestrial integration but are much simpler and thus more tractable than the large-scale heterogeneous HSTNs. In particular, we present three basic cooperative models, i.e., model X, model L, and model V, and provide a survey of the state-of-the-art technologies for each of them. We discuss future research directions towards establishing a cell-free, hierarchical, decoupled HSTN. We also outline open issues to envision an agile, smart, and secure HSTN for the sixth-generation (6G) ubiquitous Internet of Things (IoT).