A bureaucracy usually refers to non-elected officials (and their means of authority) who run a government, organization, or company. A bureaucracy is often synonymous with things like excess paperwork and red tape, meaning that a bureaucratic governing body is often characterized as having too much power which makes it difficult for the public to have any significant say in what goes on in the country, organization, or company which the bureaucracy is in charge of.
Bureaucracies are usually instituted to establish technical organization in hierarchical structures. There is nothing inherently bad about organizing a structured governing body. However, the reason the term "bureaucracy" is, in modern times, often associated with an oppressive, impenetrable governing body is that the term has come to mean the obstacles a public has to deal with under a bureaucracy that has too much power.
In Animal Farm, the Rebellion begins with good intentions. The animals take over the farm and the pigs establish a bureaucracy where they are the governing body. Had they been fair in the way they treat the other animals, their bureaucracy would have been a favorable one. However, Napoleon eventually takes over, surrounds himself with bodyguards, essentially isolating himself and the other pigs from the rest of the animals. In addition, Squealer constantly changes the seven commandments of Animalism to suit the increasing power of the pigs. These increasingly oppressive commandments (laws) are part of the bureaucratic system. That system comprises the "governors" or pigs themselves, their bodyguards, and the means of controlling and establishing rules (including commandments and labor hours).
These days, "bureaucracy" is almost always spoken of with negative connotations. Someone might refer to a government or board of directors of a company as "bureaucratic" if that government/company has a hierarchical structure that is so complex that it seems impossible for things to get done. Also, in such a bloated system, the workers or citizens have little to no means of having their voices heard. They must go through a labyrinthine complex of higher-ups and paper work that can either be daunting or useless. This is what most of the animals face with Napoleon's powerful bureaucratic system.