A litany is a long, repetitive list or series of grievances, like your picky brother's litany of complaints about dinner or the litany of critical comments your English teacher writes in the margins of your essay.
The original meaning of litany is a purely religious one. During some Christian services, a member of the clergy recites a litany, a lengthy call-and-response type of prayer. The word's Greek root means "entreaty," and in this religious context, that's an entreaty to God. Its more popular, secular meaning tends to be used in a negative way, as in your grandpa's litany of aches and pains or the litany of complaints from passengers on a stalled subway car.