The problem starts with the rapid mutation rate of the influenza virus, which causes its antigens to change.
An antigen is anything that provokes an immune system response. Flu virions are covered with proteins, which help the virus infect cells but which also act as antigens. Once "trained" to a specific antigen, the adaptive immune system is able to prevent that antigen from taking hold in the body again. This process is why a person becomes immune to most infectious diseases if they previously contracted the disease. It's also how vaccines work. A vaccine contains the antigens from a virus or bacteria, training the immune system without the need for actual infection.
With an average of over one mutation per replication cycle, both major types of flu — Influenza A and Influenza B — change rapidly in a process called antigenic drift. (Antigenic drift is different from antigenic shift, a special case of gene reassortment in Influenza A.) The antigens on the virus surface change, and the immune system fails to recognize the new antigens.
Influenza mutates so rapidly that each year, the seasonal flu consists of all-new strains. Previous flu shots are ineffective against these new strains, and a new influenza immunization tailored to the season's flu strains is necessary to prevent infection. Bird flu vaccines cannot be manufactured in advance because the specific flu strain does not even exist yet, and there is no way to predict which antigens it will have.