Impossible figures are seemingly consistent but physically impossible constructions. Generally they are two dimensional drawings of three dimensional figures where somewhere along the way something has gone awry. This is most often achieved by making the individual pieces of the figure possible while the figure as a whole is not.
One of the simplest examples is where the drawing starts out two dimensional and finishes up three dimensional.
The impossible cricket stumps are three two-dimensional pillars at the bottom and two three-dimensional pillars at the top. The standard (correct) version on the right starts three-dimensional, switches to two-dimensions and then back to three again but when it's changed back it's different. The question is where do they stop being two-dimensional and start being three-dimensional. This is a question that has no answer; hence, impossibility.
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/_/\ /0
__ \ \ \ / /
/_/\ \ \ \ / / /0
\ \ \ \ \ \ / / / /
\ \ \ \ \ \ / / / / /0
\ \ \ \ \ \ / / / / / /
\ \ \ \ \ \ or alternatively / /_/ / / /
\ \ \ \/ / the correct version \ \ \/ / /
\ \ \ / (thanks to Tiefling) \ \/ / /
\ \/ / \ / /
\ / \/_/
\/
The invisible change is another way of looking at overall inconsistency in impossible figures. The three-dimensional geometry of the impossible triangle is only possible if there is a twist in the bars that make up the triangle. A two-dimensional drawing of such a triangle does not require the twist to be drawn. Again this produces a figure that is impossible to realise.
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/ /\
/ / \
/ / /\ \
/ / /\ \ \
/ / / \ \ \
/ / / \ \ \
/ /_/______\ \ \
/____________\ \ \
\_______________\/
This is one of the less obviously impossible figures because the nature of the overall inconsistency is not obvious. Taken separately all the components of the drawing are possible. The three corners and the three bars are perfectly reasonable as the following diagram shows:
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/ /\
/ / \
/ / /\ \
/_/ /\ \ \
\_\/ \_\/
__ __
/ /\ /_/\
/ / / \ \ \
/_/ / \ \ \
\_\/ \_\/
__ __
/ /\ /_/\
/ / / \ \ \
/ /_/_ _______ _\ \ \
/______/\ /______/\ /___\ \ \
\______\/ \______\/ \______\/
The problem is that all the corners are right angles. This means the resulting triangle has 90o more than is allowed. Take a closer look at the corners. They are all orientated in different axes. Each of the corners has only one axis in common with each of the others. It is, therefore, not possible.
Take a look at the following picture. It is a perfectly normal three-dimensional cuboid. It is completely consistent. All the components are consistent, all the depth cues are consistent and its overall structure is consistent.
_______________________
/\ ___________________ \
/ \ \_________________/\ \
/ /\ \ \ / /\ \ \
/ / /\ \ \ / / /\ \ \
/ / /__\ \ \_________/ / / \ \ \
/ /_/____\ \ \ _______\/_/ \ \ \
\ \ \_____\ \ \_______/\ \ \ \ \
\ \ \ \ \ \______\_\_\_____\_\ \
\ \ \ \ \______________________\
\ \ \ / / ___________________ /
\ \ \ / / / \ \ \ / / /
\ \ \/ / / \ \ \/ / /
\ \/ / /_____________\_\/ / /
\ / /_________________\/ /
\/______________________/
To make it impossible all we have to do is to remove one of the types of consistency it needs to be real. In this case the occlusion depth cue has been messed up. The bar that is supposed to be at the back starts there but then occludes the bar that is supposed to be in front of it.
_______________________
/\ ___________________ \
/ \ \_________________/\ \
/ /\ \ \ / /\ \ \
/ / /\ \ \ / / /\ \ \
/ / /__\_\_\_________/ / / \ \ \
/ /_/_________________\/_/ \ \ \
\ \ \_________________/\ \ \ \ \
\ \ \ \ \ \______\_\_\_____\_\ \
\ \ \ \ \______________________\
\ \ \ / / ___________________ /
\ \ \ / / / \ \ \ / / /
\ \ \/ / / \ \ \/ / /
\ \/ / /_____________\_\/ / /
\ / /_________________\/ /
\/______________________/
There is another inconsistency that can be added to this diagram. The other occlusion can also be reversed to further confuse the depth cues.
_______________________
/\ ___________________ \
/ \ \_________________/\ \
/ /\ \ \ / /\ \ \
/ / /\ \ \ / / /\ \ \
/ / /__\_\_\_________/ / / \ \ \
/ /_/_________________\/_/ \ \ \
\ \ \_________________/\ \ \ \ \
\ \ \ \ \ \______\ \ \_____\_\ \
\ \ \ \ \________\ \ \_________\
\ \ \ / / ________\ \ \________/
\ \ \ / / / \ \ \ / / /
\ \ \/ / / \ \ \/ / /
\ \/ / /_____________\_\/ / /
\ / /_________________\/ /
\/______________________/
As with the triangle if we explode the diagram then it ceases to be impossible because all the components are perfectly possible on their own.
______ ________ ______
/\ ___\ /\_______\ /\____ \
/ \ \__/ \/_______/ \/___/\ \
/ /\ \_\ / /\ \_\
\/_/\/_/ / / /\/_/
\/_/
__
__ __ __ /\ \
/ /\ /\ \ /\_\ \ \ \
/ / /___ __\_\_\_____ ____/ / / \ \ \
/ /_/____\ /\___________\ /\___\/_/ \ \_\
\ \ \____/ \/___________/ \/___/\ \ \/_/
\ \_\ \ \_\ \ \_\
\/_/ \/_/ \/_/
__ __ __
/\ \ /\ \ /\ \
__ \ \ \____ __\ \ \____ _\_\ \
/\ \ \ \_____\ /\__\ \ \___\ /\_____\
\ \ \ / / ____/ \/___\ \ \__/ \/___ /
\ \ \ / / / \ \_\ / / /
\ \_\ \/_/ \/_/ \/_/
\/_/
__ __ __ __
/\ \ /\_\ /\ \ /\_\
\ \ \/ / / \ \ \/ / /
\ \/ / /___ _________ __\_\/ / /
\ / /____\ /\________\ /\____\/ /
\/_______/ \/________/ \/______/
To sum up: to make an impossible figure all you have to do is take geometric constructions that are, in themselves, consistent and link them up in a way that makes it impossible to actually create such a structure in three dimensions. The possibilities for this type of sensory deception are many and varied especially in line drawings where top can become bottom and concave can become convex with the subtlest of pencil implications.
The artist M.C. Escher, without whom any discussion of impossible figures would be not be complete, was a master of this type of optical illusion. He was the artist who popularised this type of image. It suits his style of reordering reality perfectly and if you wish to know more about impossible figures discussions of his work are the place to look. I would particularly recommend books by Bruno Ernst who was friends with Escher and has written a number of excellent books about him (see sources).
As for Escher himself the works that include impossible figures are (off the top of my head), Waterfall, Dragon, Belvedere and, Ascending and Descending. I'm sure there are many I have forgotten but this is not the node for them anyway.
Sources
Bruno Ernst, "Magic Mirror of M. C. Escher",Taschen America, LLC, ISBN: 1886155003
Bruno Ernst, "Adventures with Impossible Figures", Tarquin Publications, ISBN: 0906212545
See also more impossible figures for more of the same.
NB: These figures are impossible in Euclidean Geometry. This is not the only kind and the figures above my be perfectly possible in other geometric systems.