Cable is insulated, wire is not.
Cord usually has plugs (eg power cord to connect your PC to the mains).

I basically agree almost entirely with FJPN's answer, but wanted to add a few comments.

'cord' is originally, I think, an American usage, though seems to be increasingly creeping into UK terminology. It invariably has the sense of 'CONNECTING cable' designed/made for a specific purpose (usually with plugs on, etc. and might be for either power OR signal use)

I have to disagree with Dr G.s comment that 'cable is stiff' — of course it MAY be, but very, very many cables are also flexible (and may be described as such); it is a dangerous and inaccurate generalization to say that "cable implies stiff". The kind of 'cable' you are referring to is just one particular kind, more properly referred to as 'installation cable'. Of course there are thumping great cables, and quite dainty little ones — but the definition given by FJPN is accurate, all-embracing and quite comprehensive

The first answer is apt for electrical/electronic context. However a cord can also be akin to a string made of fibres.
A cable signifies a means of transmission or transport - could be data, signal or even people - such as in a cable car.
A wire is generally a strand of metal or metal cord or rope with the intent of connecting to points for a purpose - either to tramsnit power - electric wire or to block something between two points - barbed wire fence.

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