In larger organizations, account executives are given direct responsibility for the dealings with one or more high volume customers or clients. These customers or clients, referred to as accounts, are important to the organization's success, and so an account executive is necessary to manage one or more persons dedicated to serving the needs of the transactions and relationships with the customer.
In the advertising industry, account executives are the second-lowest-ranking members of the client service department, outranking only entry-level account coordinators. The word "executive" here means to "execute" - that is they are mostly responsible for more practical parts of the advertising jobs (ie leaflets, distributions, adaptations of materials from different languages etc). They answer to the respective account supervisor and/or to the client service director/account director. That depends on the country and on the account you are working for. In Spain for example, an account executive could probably develop very important campaigns on his/her own.