Animals rights organizations in Real Life mainly fall into three categories. Being Real Life, gray area and overlap exists between the categories. They are, in order of increasing strength:
Animal welfare groups: Usually outside this trope. These groups seek to minimize the suffering of animals, but don't object to animals being kept as pets or considered the property of their owners, and vary on opinions regarding animals as food, for testing (where the potential benefits may be sufficient to justify it), or other such uses, provided they are treated decently throughout. They just don't want animals to suffer needlessly. This site gives a good overview of the animal welfare philosophy. As a general rule, these groups are non-violent, act within the law, and genuinely care about animals and people. The only common ground between them and the groups in this trope is the concern over animals. This is generally where the ASPCA and most Humane Societies and local animal shelters fall.
Animal rights groups: Often resemble the mild versions of this trope. These groups seek to end some or all connection between animals and humans. Their goals vary from simply ending animal testing and meat-eating, to also ending the use of working or breeding animals, or circuses, with the most extreme ones seeing any ownership of animals as no better than owning humans. While they still mostly operate within the law, animal rights groups are usually a bit more provocative in their advocacy than animal welfare groups; these actions tend to be the basis for the milder fictional portrayals. PETA, discussed above, is probably the best known group of this flavour. When these groups are criticized for "valuing animals more than people" it's usually in the sense of insulting people (such as the common feminist criticism of PETA for objectifying women in advertising) rather than actually physically harming them or their property, unlike...
Animal liberation groups: Resemble the more extreme versions of this trope, except these groups exist in real life. These groups want animals "freed" from human "exploitation" and they want it right now, whatever the cost. Their means to this end can include attacks on anyone connected with, or even related to people connected with, farms (especially fur farms) and animal testing labs, and, more rarely, zoos, circuses, and the like. Here you'll find veganarchists and the ALF; the people who aren't afraid of destroying property, or harassing and threatening researchers and their families. Some of these groups are even considered terrorist groups. Wikipedia gives a good overview on the animal liberation movement. The worst ones even go so far as being Too Dumb to Live.