Gig with a hard "g" is a job. Jig, on the other hand, is a dance. The kind a band might do when they land a gig headlining Madison Square Garden.
A gig is basically freelance job. The word comes from jazz and often refers to a musical performance. If your band has a gig, you could say you're gigging. It's also short for "gigabyte," a unit of measurement in computer land. Gig likes to get around:
"Eagles of Death Metal's frontman has said he feels a deep responsibility to resume the gig which came under attack from terrorists." (BBC)
"He eventually retired from his job as an EMS for the Baltimore fire department and found a teaching gig near Hampton." (Washington Post)
"They have one job or two—or else they race from gig to gig and chore to chore as rapidly as they can." (New Yorker)
A jig has roots in traditional Irish folk dancing. Jig is also the name of the kind of song that you dance a jig to. Is a jig also a fishing lure? Sure! And, it's the name of a tool that comes in handy if you're making a jigsaw puzzle. It used to mean "trick," so if you catch someone breaking a rule, you tell 'em "the jig is up." Examples:
"It prompted Garcia to dance a jig of delight on the green." (New York Times)
"A new law goes into effect June 1 prohibiting the sale or use of lead sinkers and jigs that weigh less than an ounce." (Washington Times)
"There was also a less complex dolly jig designed to make a small doll dance." (Washington Times)
If you get them mixed up, remember that gig usually has a hard "g" for cold hard cash, and a jig makes people feel jolly.