There are many types of kart gear. First I can go over what is usually required to race.
To race, the biggest thing is a good Kart Helmet. I don't know of any places that doesn't require a helmet. And most of them require a certification rating by Snell or SFI. More about kart racing helmets on the kart helmet page.
At the very least, most places require a kart jacket, but some places may require a kart suit. Some smaller racetracks may just require long sleeve shirts or something like that. Believe me though, you probably want to invest into a nice jacket or suit to avoid burns from the engine, or scrapes if you flip an open kart over.
Some form of kart neck brace usually needs to be worn to keep your head from moving too much in any hard accidents. Most karters use the foam rings that go around your neck, but they can use other more complex forms of neck restraints.
Karting Gloves are pretty important just to protect your hands from debris and in accidents where they could get scraped up. You can buy fire proof racing gloves, but most karters just buy gloves that protect their hand from abrasions.
Kart shoes can be the high top racing shoes that are pretty thin and fire proof or just some type of high heel boot. It depends on the track you run at. It's worth investing in the shoes though, as they are more comfortable and give a better feel on the pedals.
A lot of organizations and tracks require that kids under 12 years old wear a kart rib protector or chest protector. Kids are not as strong in the mid section as adults and can bruise ribs or break them on the steering wheel in an accident.
Not required all the time, but still important are kart restraints for champ karts. Whether it is a head restraint or arm restraints, it keeps you inside the roll cage of a champ kart, because you are strapped in with kart seat belts. The seat belts are usually a 5 point harness to keep you nice and tight inside the champ kart.
Karters are very serious about safety. New ideas and kart gear comes out all the time to help racers be more safe.