Attentional modulation of cortical networks is critical for the cognitive flexibility required to process complex scenes. Current theoretical frameworks for attention are based almost exclusively on studies in visual cortex, where attentional effects are typically modest and excitatory. In contrast, attentional effects in auditory cortex can be large and suppressive. A theoretical framework for explaining attentional effects in auditory cortex is lacking, preventing a broader understanding of cortical mechanisms underlying attention. Here, we present a cortical network model of attention in primary auditory cortex (A1). A key mechanism in our network is attentional inhibitory modulation (AIM) of cortical inhibitory neurons. In this mechanism, top-down inhibitory neurons disinhibit bottom-up cortical circuits, a prominent circuit motif observed in sensory cortex. Our results reveal that the same underlying mechanisms in the AIM network can explain diverse attentional effects on both spatial and frequency tuning in A1. We find that a dominant effect of disinhibition on cortical tuning is suppressive, consistent with experimental observations. Functionally, the AIM network may play a key role in solving the cocktail party problem. We demonstrate how attention can guide the AIM network to monitor an acoustic scene, select a specific target, or switch to a different target, providing flexible outputs for solving the cocktail party problem.