If you've bough a fully assembled and tested scooter from a local dealer than they should take care of all the required registration. All you have to do is pay the dealer and your fully registered and inspected scooter should be delivered to you.
However if you've bought a mail order chinese scooter from out of state you'll have some work to do. I can only describe what's required in New Jersey, though I suspect that most states have similar requirements. The one exception may be California where vehicles have to meet stricter emission standards than other states (CARB) and scooters that can be registered in the other 49 states may not be legal in California. That's somewhat ironic because most of the chinese scooter importers are actually located in California.
In NJ you'll need a manufacturer's statement of origin (MSO), which the dealer should provide. This is a certificate from the scooter manufacturer which has the make, model and VIN (Vehicle Identification Number) on it and is proof that the scooter meets all applicable DOT and EPA standards for registration in the USA, and in particular for the state you are trying to register it in. For NJ the following link lists currently approved motorcycle manufacturers. If your scooter manufacturer isn't on this list, registering it is going to be an even bigger pain than if it is! Here's the link http://www.state.nj.us/mvc/pdf/Vehicles/mc_manufacturers.pdf. You also need proof of insurance, the dealer invoice and a valid driver's license. Then you simply drive to your local DMV office, wait in line for 20 minutes, get the forms, fill them out, wait in line for another 30 minutes and if all your documents are in order you get a title and your motorcycle plates (moped plates if your scooter is under 50cc).
Then you have to take your scooter to an inspection station (within 14 days for NJ) along with a yellow card which the DMV office provided, proof of insurance and a valid drivers license (with motorcycle endorsement if the scooter is over 50cc) and the inspection station will give you an inspection sticker good for 4 years. They may also check that your helmet meets local requirements (DOT certification etc).
Then you are good to go! Licensed, registered, insured and inspected.
However beware of problems with the MSO. Every approved manufacturer is given World Manufacturer Identifier or WMI code, which is used as the first three characters of the VIN number. If the first three characters of the VIN don't correspond to the manufacturer on the MSO, the DMV won't register the scooter. Believe me, they won't because I went through that. The MSO certificate I was sent has the wrong manufacturer's name. I was told it was a Roketa and the invoice said Roketa and the MSO said Roketa. However the VIN number had the WIN for JMstar and the plate on the scooter said the scooter was made my JMstar. It took three weeks and several phone calls and a 2 hour visit to the DMV to get a corrected MSO issued by the dealer and register the scooter (the VIN had been flagged as "bad", so even with the new MSO the DMV had issues which took an hour or more to sort out). Dealing with the DMV is no fun.