Solicitation is a request for something, usually money. If you buy a fancy new house, expect a phone call from the local charity with a solicitation for a donation.
Solicitation comes from solicit, which means "to request," or "to entreat." So solicitation is the act of requesting. There are three kinds of solicitation. One is asking for money, like when someone goes door-to-door trying to collect money for a cause. In law, solicitation means encouraging someone to commit a crime. Solicitation can also mean making a plea to someone with power to grant a request or favor, like your solicitation to the traffic judge to dismiss your ticket.