The Questing Beast is introduced in chapters 19 and 20 of the first book; note that different editions may have different chapters. It is mentioned in chapter 12 of book 9, and strangely, it is better described here than in its initial appearance but doesn't actually appear in the events described. It is mentioned in several subsequent books but usually in little more detail than to be named.
In book 1, Arthur encounters the Beast while resting in the woods, noting the Beast's unusual sounds, which are akin to the sound of thirty dogs hunting. He then meets King Pellinore, who requests the use of Arthur's horse in order to continue his pursuit of the Beast. Arthur protests that he might keep the horse and simply take up the pursuit of the Beast, an offer that Pellinore discards somewhat callously, claiming that neither of them is destined to actually catch the Beast in their lifetime.
The Beast isn't particularly relevant to the story and is almost entirely inconsequential in that it appears, Arthur remarks upon it, and from then on, it exists simply as a background detail. One might assume that there must be some deeper meaning, such as symbolism or a variation on a theme, in order to justify the Beast's presence in the story, and indeed there is considerable scholarly research on the Beast's literary and folkloric origins and the purpose it serves in the text.