For the previous parts in this series, see the Introduction, Ad Hominem, Strawman, Begging the Question, and Slippery Slope articles.
The fallacy of equivocation is essentially made when you use a term in the premises in your argument in two different ways. For example, take the following argument from the linked webpage:
P1: Brad is a nobody.
P2: Nobody is perfect.
C: Therefore, Brad is perfect.
The fallacy here is apparent: In the first premise, you are using the word "nobody" to mean "someone unknown." In premise two, you mean "nobody" to mean "absolutely no one." You are switching terms illegitimately to prove your conclusion.
This is a fallacy of ambiguity. In order to make a convincing logical argument, you must use all the terms in the same way in all of your premises and make your arguments as specific as possible.
Examples of equivocation in the abortion issue:
A pro-choice equivocation I hear very often goes something like this: "Human beings have basic rights. Fetuses cannot drive or vote. (From 1 and 2) Therefore fetuses do not have basic rights. (From 1 and 3) Therefore fetuses are not human."
This equivocates on the term rights. The person making this argument is failing to make a distinction between natural rights, rights that human beings have by virtue of being human (the rights to life, liberty, the pursuit of happiness, bodily autonomy/integrity, etc. fall into this category) and legal rights, rights that the government bestows on its citizens and can rightly take them away (the rights to vote, drink, drive, etc. fall into this category). So the equivocation is obvious here: The first premise talks about natural rights, but the pro-choice advocate appeals to legal rights in their second premise to lead to their conclusion.
Pro-life people tend to be accused of equivocation with the standard argument, that it is wrong to kill innocent human beings, fetuses are innocent human beings, therefore it is wrong to kill fetuses. Mary Anne Warren, in her essay On the Moral and Legal Status of Abortion, made this charge fairly famous. The charge is that in the first premise, the genetic (biological) sense is in mind, and in the second premise, the moral sense is in mind. However, this is a mistaken charge. The term "human beings" is used in the same sense in both premises, as a full-fledged member of the human species. One could mean the genetic sense in both premises and arrive at the same conclusion without equivocating. Then Peter Singer may accuse you of speciesism, but of course, it is not wrong to hold that the human species is special among all the animals on Earth because it is not merely belonging to the biological species, but the kind of thing that species is, that grounds their value. Human beings, all human beings, have the inherent nature as rational, moral agents, and that grounds their value.