Olympic Bar vs Standard Bar
What you'll see right away is olympic bars are thicker on the ends, because they have a sleeve that rotates over the shaft. But that's just one of many differences when you take a closer look at them in person.
Standard bars are exactly 1″ in diameter throughout the whole length (besides the shoulders that stop the weight plates). They are made to take standard weight plates.
Standard bars weigh 13 to 19 lbs or so. That's for solid steel ones. Special "cardio" bars are hollow, only 5 lbs, and are not meant to take much weight.
Olympic bars are just over 1″ diameter in the center shaft area, or just a tad larger than standard bars, and then they are nearly 2″ diameter on the ends (actually 1 31/32″, or 50mm). As such, they will only take olympic weight plates, which have 2″ diameter holes. These actually can also be divided up into olympic weightlifting and powerlifting bars, or even further, but they are often just called olympic bars.
Olympic 7ft bars are supposed to weight 45 lbs or 44 lbs (20kg). There are cheaper 7ft bars that weigh less and are very low quality. Shorter olympic bars, 5ft or 6ft, may weigh as little as 29 lbs. "Technique" bars are hollow or aluminum and a good choice for sticking with olympic weights but at a low starting weight as low as 10 lbs.
One of our most popular guides presents the best olympic bars under $200.
BWTB also has this blog post on barbells from a Crossfit perspective that you should check out.
Specialty Bars
While there are standard curl bars and a couple other types, most specialty bars like the Rogue multi grip bar shown below are only found in olympic sizes. There is much more demand for specialty bars from the serious weight training crowd who only use olympic weights.
Both types can come in a variety of styles.
Both types are generally all made of the same cast iron, other than the solid rubber olympic bumper plates, and then covered with a baked-on enamel paint or a rubber or urethane coating.
Typical standard plates are the "pancake" shaped plates with rounded edges.
Standard plates consist these sizes: 1.25lb, 2.5lb, 5lb, 10lb, 12.5lb, 20lb, 25lb, 50lb. The center holes are just over 1″ diameter to fit on a standard bar.
Olympic plates come in more styles, like the below "grip plates", ie: with holes to help you grip for easier and safer handling.