Transposition is simply moving the relative positions of letters within a message. I'll describe a columnar transposition below, so called because the text is arranged into columns and the columns are transposed.
When performing a columnar transposition we first need a keyword. I'll use the keyword murkydotorg. The first step is to remove duplicate letters from the keyword, this yields murkydotg.
The message is then written into rows beneath the keyword. The example message which I'll use will be you can read murky dot org via the convenient RSS feeds
MURKYDOTG 476381572 --------- youcanrea dmurkydot orgviathe convenien trssfeeds
You will notice that I've added some numbers beneath the keyword. The numbers refer to the relative positions of the keyword letters in the alphabet.
Having formed the table we can read back the message in the order of the keyword letters.
This message becomes NYANE ATENS CRVVS YDOCT RDTIE UUGNS OMROT AKIEF. I have included the spaces for clarity, although in practice these won't be highlighted!
One should also note that there are many ways of systematically anagramming the letters of the message. I am sure that you can come up with some methods yourself!
One well known method is 'railfence', where the letters are written in a zigzag, and read off in rows. The height of the zigzag, as well as starting direction and starting row can be adjusted.
n u t a n n e y a r m r o o i t o v e t f e o c e d k d r v h c e i r s d u a y g e n s s
This becomes: nutanneyarmrooitovetfeocedkdrvhceirsduaygenss
Transpositions are often used as part of a more complex system. If a transposition is used in conjunction with a monoalphabetic substitution then we may solve the transposition as above, after having first worked out the plaintext letters of the substitution by looking at letter frequencies. Quite often though, a transposition is solved by much trial and error!
When solving transpositions, one might have fingers crossed that letters like 'q' appear, as this is (usually) followed by 'u' - some exceptions exist, like qwerty and qed.
Similarly, sometimes 'chunks' of text are transposed, so a large message may be transposed in blocks of 25 letters, and solving one block will solve them all.
It should also be seen that, as with many cryptographic systems the more ciphertext we have, the easier the cipher is to solve.
Transposition add some security, however as with all cryptography, one should be aware that sometimes the solution is easier than we might lead ourselves to believe. Personally, I don't like trying to solve transpositions - but this is a matter of personal taste.