To put it in terms that anybody who has learned a foreign language can understand, declension (from Latin declinare, to turn aside) is analogous to conjugation, except that it works on nouns. Just as conjugation gives information about who is performing the action, how many of them are there, how they are doing it, when it happened, declension gives information about how the noun stands in relation to other nouns, which it is connected to, which verbs are taking it as an object, and more.
Arguably, the canonical declined language is Latin, due to its position as the declined language which more natives and scholars have learned than any other. Latin has five separate declensions, each with seven cases each in singular and plural: the nominative, which names the subject; the genitive, which expresses possession or partitivity; the dative, which is used for indirect objects and a smattering of specialized adjectives; the accusative, which is used for direct objects, as well as following some prepositions; the ablative, which follows other prepositions and can be used to express how or when something happened; the vocative, which is used in addressing people; and the locative, used to express motion towards a place.
A major difference between a declined language and a nondeclined language is that word order is arguably less important in a declined language. Because I can say mihi eam duc or duc mihi eam or any of the other four possibilities and still be saying lead her to me, sentence structure can, in some cases, go straight out the window. However, in English, I cannot say her lead me to, lead me to her, or even (with more complex phrases) something along the lines of to me her lead. The abuses of a strongly declined language are well-known to any student who has had to read Virgil, who was very fond of writing enormous, six-line sentences with all the verbs saved until the very end.
Other cases of declension, such as the instrumental, the comitative, the superessive, and at least 30 others (linguistics is a complex field) appear in other languages. In the case of Latin, several present in earlier languages were subsumed into the ablative case as the language developed.
Other declined languages include: (thanks to izubachi for aid)
For more information on declension, see Monty Python's Life of Brian.