Garrett (c.1895) points out what a difficult task it is to define the meaning of this word:
The term itself, which, according to Skeat, is of Celtic origin, is in culinary parlance extended so widely by the fancies and tastes of cooks that it is difficult to assign any limitation to its application. Webster describes a Pudding as a species of food of a soft or moderately hard consistency variously made, and this we are compelled to accept, having nothing more definite to offer.
Nonetheless, people constantly try to pin down a more exact meaning. The NSOED gives:
A cooked dish consisting of various sweet or savoury ingredients, esp. as enclosed within a flour-based crust or mixed with flour, eggs, etc., and boiled or steamed; a baked batter mixture. Now also, the sweet course of a meal.