If it's a consumer gadget/toy (and not a medical device which has much more stringent standards!) you probably just need to worry about safety standards and EMC standards.
- CE certification for Europe
- EMC certification: in the USA, FCC class B radiated emissions
- Europe now requires RoHS to limit Cd, Pb, Hg and some other hazardous substances
Safety standards will include things like pinch points, sharp edges; if it's a toy and there are children involved, there's a standard container used for choke hazards, etc. If the gadget plugs into the wall for power, you need to worry about shock hazards and fire hazards (UL in the USA), although you can probably deal with this by using a UL/CE approved wall transformer so voltages entering your device are limited.