A palindrome may be a word, phrase, number and/or any other type of sequence of units that can be read with the same result in either direction (capitalization, punctuation, and spaces between words are typically ignored in qualifying something as a palindrome, and different diacritical marks on matching letters normally allowed). Many more details may be found at the Wikipedia entry for Palindromes..
The English word "palindrome" is thought to have been coined from the Greek roots πάλιν (palin) meaning "again" and δρóμος (dromos) meaning "way, direction" by English writer Ben Jonson in the 1600s. The actual Greek phrase to describe the phenomenon is καρκινική επιγραφή (karkinikê epigrafê) meaning crab (-like) inscription, or simply καρκινιήοι (karkiniêoi) simply meaning "crabs", alluding to the skittering side to side, backward and forward movement of crabs, since the inscription can be read forwards and backwards, or side to side depending on the way you look at it.
Composing literature in palindromes is an example of Constrained Writing. Palindromes are a particular type of Constrained Writing, i.e.: a type of writing where the writer imposes a literary technique in which they are bound by certain conditions that require and/or forbid utilization of certain components, forms, and/or techniques, and/or imposes a particular patterning on the work. The Wikipedia article on Constrained Writing, contains much detail and is well worth checking out for those interested.
While most definitions of palindromes concentrate primarily on the ability to read the same message in both directions, it is secondarily important within the multi-word or phrase type of palindrome is that the all the components be actual words, or in some cases idiomatic contractions. The best palindromes usually contain several nuanced layers of meaning. Metaphor often plays an important part in palindromes, especially those which become utilized in Spiritual or religious contexts such as chants, or other Ṇ't̄ā
This Palindrome is used in chants often dating back to various archaic Ægean cultures:
(transliterated as: Nipson Anomimata Mi Monan Opsin)
[rough meaning: I wash away the actual problematic behaviours as I wash my face]
Thus being ideal for utilization during the cleansings in preparation for Werkings of various sorts.
Note: it is a Palindrome in the orginal Greek since the "pS" sound is represented by a single letter, it doesn't work out the same when transliterated to English where they are two.
Another common metaphysical utilization of Palindromes is in Palindromic Word Squares, which ideally are able to be read in all four directions. The most famous of these appears to be the Latin word square "Sator Arepo Tenet Opera Rotas" was found as a graffito at Herculaneum, buried by ash in that year (79c.e.). This palindrome is a special case, remarkable for the fact that it also reproduces itself if one forms a word from the first letters, then the second letters and so forth. Many translations have been offered and used, the Arepo word being the most problematic.
Ambigrams are different from Palindromes, but they have some similar attributes. An Ambigram is an illustration or some other sort of typographical device which makes a graphical representation of a word readable from different points of view as either the same word or a different word. In the recent Ron Howard film of the Dan Brown novel Angels and Demons, the ambigram of Illuminatus reads by calligraphic modification, as Illuminatus both forwards and backwards, implying the omnipresence and umbiguity of the Illuminatus as it is commonly conceived. Of course the Illuminati is the catch-all name for everyone's favorite universal conspiracy secret society, though the likelyhood seems extreme that there are multiple Illuminatis (one would think at least five, for symmetry) due to reasons elaborately gone into in Robert Anton Wilson and Robert Shea's immortal Illuminatus Trilogy:
The above image is an Ambigram, and obviously not a Palindrome, though there are "Illuminatiesque" phrases such as,
"No Omega drags an Illuminati Titan. I mull in Asgard, age Moon."
and the like which are actual Palindromes.
Able was I ere I saw Elba
Was it a cat I saw
No Valiant at nail Avon