Wealthy and underpopulated by Central African standards, which is not admittedly saying very much, Gabon is positively placid compared to neighbors like DR Congo.
Health: Malaria and HIV/AIDS are the main dangers. It's host to the full gamut of tropical diseases, including Ebola (not the current 2014 outbreak though), but you're unlikely to run into these unless you really go all-out in the jungle.
Travel: Roads and driving standards are horrendous. At least you've got the Transgabonais train as an alternative for some places, eg it'll take you most of the way from Libreville to the Congolese border.
Crime: Treat Libreville like any other large African city, so keep your wits about you, don't flash valuables, avoid travel alone at night, etc.
Corruption: Carry your passport at all times. Pretend not to understand requests for bribes.
Civil unrest: No civil wars, significant rebel movements etc. Occasional demonstrations against the Bongo du jour.