Everything depends on the source of the zombification. In early zombie lore a person was raised from the dead by a medicine man and did his bidding, so it could attack and kill people but may or may not eat them, and didn't create other zombies.
In Romero films and assorted media, the source is hinted at being some sort of alien radiation. Some zombies are hungry and eating human flesh, others are just malevolent and want to kill people and use tools to do so. They don't always leave enough of a person left to turn but when they do, that person is infected.
In fast zombie revival films, the zombies appear to have some sort of rabies-like virus that make them self-destructive as well as other-people-destructive, and staying intact long enough to turn yourself depends on how many zombies are attacking you. They seem to give up and run off to find a new victim when the old one stops moving, so a lone zombie will attack you until you bleed out and then feck off to attack someone else.
In the Walking Dead and other related media, everyone is infected (airborne pathogen?) but living people are asymptomatic, it only reanimates once the person is dead, and getting bitten gives you a dose of the active virus that kills you. The virus probably repels maggots that would normally eat the flesh.
The real-life fungus Cordyceps is the inspiration for The Last of Us and a few other zombie-related media. Its spores infect insects and caterpillars and grow over the brain, influencing the affected creature to do things beneficial to the fungus and eventually killing it and feeding on its corpse.
Other misc sources are self-replicating nanomachines, demonic possession, evolutionary kill switches, brain parasites and voodoo curses.