A first-order (or SN1) substitution reaction occurs when one group on an organic molecule leaves and is later substituted by another group. This reaction goes through a carbocation intermediate. The SN1 mechanism is analogous to breaking up with your current significant other, staying single for a while, and only after you've been single, becoming attached to a new romantic partner.
The likelihood of an SN1 reaction occurring depends on the solvent, the nature of the substrate, and the substituting group (called the nucleophile, or nucleus lover, sometimes abbreviated Nuc). As the name SN1 indicates, it requires a Substitution by a Nucleophile in which 1 reactant determines the rate of the reaction. The SN1 mechanism is shown here.
As you see in the figure, the SN1 reaction proceeds in two steps. In the first step, the leaving group decides to pack its bags and take off, leaving a carbocation intermediate. Then the nucleophile attacks the carbocation to make the substituted product by replacing the leaving group.
The nucleophile can be weak. However, the reaction does require a weak base as the leaving group.
The substrate and solvent are also relevant. A tertiary substrate (where the carbon is bonded to three other carbons) makes second-order reactions extremely difficult, and so it is more likely to favor first-order reactions. A polar protic solvent also favors SN1 reactions.
Protic solvents contain O-H or N-H bonds.