'Third party' is the minimum insurance you must have to drive a car on UK roads. It covers you hitting someone else. However this does not cover any damage to your car if you are at fault.
'Third party, fire and theft' is the next stage up for us, increasing, only protects you if the car bursts into flame or gets purloined by a suspicious person for nefarious purposes.
'Fully comprehensive' or 'fully comp' is the one most people go for if they can afford it. It covers you if you are at fault, and covers you if something has happened to your car but you cannot trace the miscreant for a damn good thrashing, err fiscally punitive legal enforceable penalty, ie getting the [expletive deleted] to pay for the damage they have caused.
I've been fully comp since before I could afford to be fully comp without sacrificing something else.
Depending on the excess required, and cost differential, fully comp equivalent is what you are after, if unavailable, fire and theft.