Papyrus is Sans's younger brother. Despite their difference of age (5-6 ages), he's greatly taller than his brother, more serious and responsible.
Like Sans, Papyrus takes place in the mission for fallen humans catching, but he does this because of his ambitions. The truth is that this skeleton can't use genjutsu or ninjutsu, and he wants demonstrate that if you use only taijutsu, you can be the true shinobi too (yes, he's a bit like Rock Lee because I think that these great and lovely characters have some similarity =^_^= ). If he catches a human, other monsters will find him like a great shinobi, will respect him and, how Papyrus thinks, he will have more friends. Yes, in Snow Village he's very lonely, nobody takes him seriously and his ambitions sound quite funny for a lot of monsters.
Nevertheless, Papyrus doesn't lose heart and, having whole-hearted brother's support, hardly trains everyday. Undyne, leader of ANBU, notices his pains and, being imbued with his determination end enthusiasm, decides to help him with training. Despite few monsters have admited him, as a result he's stronger physically than his brother, according to Sans's words (Sans isn't very powerful in taijutsu, his benefits are always genjutsu and ninjutsu, like a opposition to his brother).
For this moment (maybe I'll change this later if I imagine something better), the plot of story after leaving the Ruins is like that: Sans, having met Frisk, brings her in Snow Village, having hidden her from ANBU. At home he's asking kunoichi about ninja on Surface, why she has climbed on Ebbot, and what she is going to do. Also, Sans agrees to teach and train Frisk. Hovewer, their conversation is interrupted with Papyrus's arriving at home. Nevertheless Paps has never seen humans in his live, he understands who is Frisk. He's trying to catch her, but she flees. Being in Snow Forest and hidding from determined Papyrus, Frisk runs up against ANBU's dogs (sometimes with Sans's help, of course), gets away from younger skeleton's puzzles (which are more like traps), and in the end she goes into battle against Papyrus (I'll tell about this fight in more details sometime).
And a bit more about skeleton: he's very optimistic, punctual, friendly and true friend, but sometime with high self-conceit. Papyrus wears a form-fitting and cosy clothes which don't rivet him in battles (but Sans often mocks him about his clothes because they are too form-fitting)). Also, in battles he sometimes uses long bones with sharp ends which are like daggers. He is very good at cooking, Paps's favourite dish — ramen, but his recipe is a little bit different from the best recipe of Snow Village's cafe — "Grillby's Ramen".
The next reference is about Grillby and Fuku Fire (maybe. or not.).