A semi-automatic gearbox is an automatic box with manual override. The on-board computer systems combine data from the engine, transmission and gear lever to control the throttle, clutch and gearshift during each change. The driver stays on the throttle during the shift and merely pulls the lever to go up a gear or pushes it to go down, the clutch pedal is not needed except for pulling away from rest.
A semi automatic gearbox is a manual gearbox with an automatic clutch. So that there is no clutch pedal. This means that by pushing the gear lever or paddle in one direction it will change up a gear - the other direction changes down. Very often there is the facility to drive in an automatic mode. The gearbox on a semi automatic is the same as for a manual - but the clutch works automatically
I own a semi auto renault megane. My car has an ordinary gear box as you would find in a manual car and gearshifts are completed in the same way as a manual, although i have been told (by my mechanic who owned one), there is an older style saab which has the same style gearbox will actually change the gears for you if you forget.