A person who has a low tolerance for alcohol and becomes intoxicated quickly after relatively few drinks.
The expression originates from the "traditional" dating scene where men pay the bill for the evening, including for the woman's drinks. The intent is that little money is needed for them to either have had their fill for the evening, or to be drunk enough that sex is likely to follow.
In modern times, the term is applied almost equally to men and women regardless of orientation.
"Frank sure is blitzed."
"He's only had two. Must be a cheap date."
A person, usually a girl, who you can take out without worrying about $100 restaurant tabs. She's usually a vegetarian or prefers pasta and appetizers to a steak and lobster dinner. Some prefer going to a cafe or fast food instead of a fancy restaurant. If she doesn't drink any alcohol that's even cheaper.
Barbara chose pizza when we went out Saturday night and bowling the previous week. I thought she would be the type to break the bank but so far she is a cheap date.