It seized. Happens to two strokes quite a bit. Luckily new cylinders are cheap and easy to put on for any home mechanic. There are 4 bolts that hold the head on the cylinder. Pull the head and check for scoring along the cylinder walls. If good I would change the piston ring (rings) and call it a day. You may also check to see if the carburetor is jetted properly. If it was fuel starved it will seize because the fuel cools the engine as well as air.

Its unlikely its seized or needs a new barrel and piston if it runs ok when its going. Does it have a fuel filter? Look at that. Its likely to be electrical and you need a new coil as that explains the symptoms. Simple things first.

It might of seized, due to low or no oil. or it died because the gas tank is not vented. Next time try opening the gas tank filler and see if it starts up after a few cranks.

2 things to do:

1. Replace the airfilter (you can get these from the internet or your local garage)(they'll fit it too)

And 2. Replace the spark plugs. (Again, mechanic)

I bought my girl a used 50cc jetstar scooter, today on the way to work it just suddenly lost power (motor not electric) and died. It started after a few minutes then did the same thing, after letting it cool for like 10 min it started up and i rode it home, any advice as im not used to these small engines