The goal of doublets is to turn one word into another by changing one letter at a time; each of the linking words must also be a word. Or as Carroll explains:
"The rules of the Puzzle are simple enough. Two words are proposed, of the same length; and the Puzzle consists in linking these together by interposing other words, each of which shall differ from the next word in one letter only. That is to say, one letter may be changed in one of the given words, then one letter in the word so obtained, and so on, till we arrive at the other given word. The letters must not be interchanged among themselves, but each must keep to its own place. As an example, the word 'head' may be changed into 'tail' by interposing the words 'heal, teal, tell, tall'. I call the given words 'a Doublet', the interposed words 'Links', and the entire series 'a Chain', of which I here append an example:
TAILIt is, perhaps, needless to state that it is de rigueur that the links should be English words, such as might be used in good society."
It's a fairly simple concept, and Carroll himself acknowledges that"I am told there is an American game involving a similar principle." But it is a game with enduring appeal, as evidenced not only by the Electric Company, but by its inclusion in modern puzzles compendia, such as Games Magazine (which I also get), where it often described as "word ladders."
You can read the whole Doublets book here, courtesy of Google Books and test your brain power with the puzzles he proposes. Carroll himself notes that the easiest doublets are those in which the position of vowels and consonants is stable, so you just have to swap vowel for vowel or consonant for consonant. Transforming vowels into consonants and vice versa is much trickier.
As a parting note, there was also a very nice article on Carroll in this month's issue of Smithsonian Magazine (yet another one I subscribe to). It focuses on his relationships with children and the various ways that has been treated by his biographers over time.