To "give someone the slip" is to escape, or get away from that person. Example: "Quick- there is a police car behind us; let's give him the slip!" Something "slips" (verb) when it gets away from where it is supposed to be. A "slip" (noun) is an old English word for a fake coin; if you bought something and paid for it with a fake coin, you gave the seller "the slip". To do something to get away from someone is to "give him the slip." Example: "My brother will be at the movie tonight. Afterwards, let's give him the slip and go to a party." The moment you got away from that person is the moment you gave him the slip.