When and why did the Catholic Church first start issuing annulments? I understand that the Catholic Church issues annulments in place of divorces under certain circumstances.  When did this practice first begin and under what was the reasoning given for it and its distinction from divorce?
Definition: An annulment doesn't dissolve a marriage; an annulment is a solemn pronouncement that there never was a marriage contract (as explained in the pronouncement) and, therefore, what was perceived as a marriage was in fact null.

Why: In order for a marriage to be valid, it cannot be null.  If a marriage is ruled to be null then the couple can either try to fix the issues regarding the cause of nullity or must separate (which typically already happened).

When: Presumably this practice was with the Church from the beginning. The reason it was so rare previously is that Catholics used to be taught their Faith before seeking to get married, and as part of their marriage preparations they used to be "annulment proofed" both by the priest insuring the couple are mature, sane, morally capable and aware adults, as well as the custom of "reading the bands" which, among other things, was a moral requirement for anyone who knew of an impediment for the couple to get married (which would be grounds for nullifying the attempted wedding contract) to come forward and make it known to the priest.  

These days, with inadequate education in the Faith, general immaturity of the populace in general and people seeking to get married in particular, and no moral requirement to make impediments known to the clergy, it can only follow that the number of annulment cases will skyrocket.