For the strong wire with sharply pointed barbs, barbed wire is the standard term. It’s simple: barbed is an adjective modifying the noun wire. The variants barb wire and barbwire are listed in some dictionaries and appear in quoted speech and informal writing, but barbed wire is more logical and is also more conventional in the industries that produce and use the wire.
Bobwire, bob wire, and bobbed wire do reflect how some people pronounce the term, but those spellings are widely considered incorrect.
As a noun phrase, barbed wire is two words with no hyphen. As a phrasal adjective preceding the noun it modifies, barbed wire is hyphenated—for example, barbed-wire fence.
Example
In the following examples, barbed wire is a noun phrase and hence unhyphenated:
They’re tucked into the side of a hill and surrounded by huge sandbags and ribbons of barbed wire. [NPR]
Barbed wire, weapons and gas-filled bottles are all very useful along with alarms. [Guardian]
A Great Horned Owl is recovering at a Nanaimo veterinary clinic after it was rescued from barbed wire. [Canada.com]
And in these examples, barbed wire is hyphenated because it is a phrasal adjective:
She pushed through a gap in the barbed-wire fence, darting in front of vehicles and across the busy street into nearby woods. [Boston Globe]
The grasslands surrounding the Darlington Nuclear plant are flanked by barbed-wire fencing and regularly patrolled by armed guards. [National Post]
Most of them still live on Escobar’s former property in a lake surrounded by a flimsy barbed-wire fence. [Sydney Morning Herald]