Jigs
A jighead can be used to rig a soft plastic lure or to use with natural bait. Typically you tie the jighead to your leader material with a tight or loose knot, and then connect your leader to your mainline with either a line-to-line knot or a swivel. This is one of the most versatile rigs you use. You can switch from dead shrimp, to live mullet, or to a soft plastic lure in just a moment.
Weighted Hooks
These are for rigging soft plastic lures in a weedless arrangement. This allows you to fish docks or oyster beds where snagging is a problem. They would be rigged like the jighead above.
Rigging Soft Plastics
This video is an excellent resource for these rigs. It includes a link to a PDF cheat sheet.
Dropper Style Bottom Rig - Fancy - Simple
This is the rig for surf fishing. You can make as many droppers as you like. You can use little floats or not. You can use swivel hardware or just knots.
Carolina Rig - Simple
Note: The less metal, the better. Snap swivels and wire leaders are very unnatural. Simple mono is best.