The example supposes that a mad scientist has removed your brain, and placed it into a vat of liquid to keep it alive and active. The scientist has also connected your brain to a powerful computer, which sends neurological signals to the brain in the way the brain normally receives them. Thus, the computer is able to send your brain data to fool you into believing that you are still walking around in your body.
The brain-in-a-vat thought experiment is generally used to ask the question: how do you know that you are not a brain in a vat? The question mirrors an early one from Descartes, which asks how you are to know that there is not an evil demon feeding false information to your senses. The essential conclusion is that, from the perspective of the brain itself, it is impossible to tell whether it is a brain in a vat or a brain in a skull.
Putnam himself uses the example to argue for a truth theory that essentially side-steps the problem of scepticism. Putnam says that even if you are a brain in a vat, you should count the things you believe that you experience as knowledge. For example, if you say that you have a nose, but you are actually a brain in a vat then you are correct, but what you are actually referring to is your virtual nose. The proposition which you state is true in the sense that it is a short-hand for what is epistemically available.
Putnam’s answer to scepticism has been met with the criticism that it ignores the problem of scepticism by merely stating that potentially false things will count as true for apparently pragmatic purposes, and that it essentially forbids assigning truth values to statements about things that may be outside of the epistemically available world. For example, a brain that is really in a vat could never truly assert that it is a brain in a vat, or talk about anything in the “real” world.