The idea that corporations should be afforded the same "inalienable rights" as people.  In America, this came about in 1886 when the best and brightest US plutocrats convinced the courts that corporations should qualify for equal protection as promised people who are citizens of the US by the 14th Amendment of the US Constitution.

The clause in question runs:

All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the state wherein they reside. No state shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any state deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws.

Emphasis mine.  Since corporations qualify for 14th Amendment protection, they get all the good stuff like free speech, due process, etc.

Some folks will tell you that corporate personhood is a bad thing.  They have lots of crazy reasons, like

  1. Corporations don't have the same responsibilities as people.  While they pay taxes, they can't be drafted, can't be punished in the same manner as people if they break the law (that is, if a corporation is found guilty of murder in Texas, it can't be put to death), etc. etc. etc.  Since corporations don't have the same responsibilities as us, they shouldn't have the same rights.
  2. Giving corporations 14th Amendment protection actually breaks equal protection.  Since corporations have more resources than people, they can protect their rights where people can't.  In practice, then, they actually have more rights than noncorporate persons.  For example, corporations can afford lots of fancy lawyers whereas most folks can't afford to go to court.  Consequently, if you give corporations the same rights as people de jure, they will have greater rights de facto.
  3. The point of protecting rights is to make life better.  Corporations, by their nature, are disposed to value profit above all else, even when it means that people lose their rights or die.  Decision-makers in the corporation can, f'r instance, be sued if they fail to maximize profit for their shareholders.  Yes, they can actually be punished for making decisions like "Drill for oil in the Alaskan Wildlife Refuge?  No way!  Let's look elsewhere."  Corporations, therefore, need to be rigidly controlled to protect the rights and lives of human beings since it has been emphatically demonstrated time and time again that they will do terrible things if left unconstrained.
People who make these arguments, however, are obviously moron commie nutbags. Reasonable people realize that the only way for anyone to be happy is total laissez-faire capitalism.  After all, you can't make an omelet without killing a few people.