An S1 reaction occurs when a molecule separates into a positively charged component and a negatively charged component.  This generally occurs in highly polar solvents through a process called solvolysis.  The positively charged component then reacts with a nucleophile forming a new compound. S1 reactions are reactions whose rate is dependent only on haloalkane concentration.
In the first stage of this reaction (solvolysis), the C-L bond breaks and both electrons from that bond join LG (the leaving group) to form LG and RC ions. This is represented by the curved arrow pointing away from the C-LG bond and towards LG. The nucleophile Nu, being attracted to the RC, then donates a pair of electrons forming a new C-Nu bond.
Because an S1 reaction proceeds with the Substitution of a leaving group with a Nucleophile, the S designation is used.  Because the initial solvolysis step in this reaction involves a single molecule dissociating from its leaving group, the initial stage of this process is considered a uni-molecular reaction.  The involvement of only 1 species in the initial phase of the reaction enhances the mechanistic designation to S1. An S1 reaction has two steps.