What is this thing? It was shown in Jabba's Palace, sitting with Jabba, and seemed like it was probably some sort of favored pet.
Is it sentient? Does it have a name, or an identified species?
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That is Salacious Crumb. To quote the Star Wars Wiki:
Salacious B. Crumb was a Kowakian monkey-lizard employed as a court jester for Hutt crime lord Jabba Desilijic Tiure.
And yes, Jabba the Hutt has a full name.
Regarding their sentience, the Star Wars Wiki says:
Although they had no established culture, Kowakian monkey-lizards were quite intelligent, and despite their silly, cruel nature, they were considered sentient by many sentientologists in the galaxy, although there was much debate over their possible sentience.