Infinitives and gerunds are verbs that are changed into nouns. Infinitives are verbs with "to" added, like "to study" or "to drink." Gerunds are verbs ending in "-ing," like "studying" or "drinking." The document provides examples of when to use infinitives and gerunds after verbs like "like" and when something "is fun." It also gives examples of common mistakes in using infinitives versus gerunds.