The wheelbarrow was not originally invented in a gardening context. The first wheelbarrows were invented in China for the transportation of military supplies.
The wheel barrow is a second-class lever. The wheel is the fulcrum. The input force is at the handles and the output force is the load in between. Wheelbarrows have sloped sides to facilitate tipping out the contents of the barrow, either out the front or to the side.
Metal barrows are more durable and usually more aesthetic but also heavier and noisier than plastic wheelbarrows. If you are a serious gardener, you will probably want a light barrow for weeds and a heavy-duty barrow with a reinforced undercarriage for heavy loads - it will not bend when used to transport soil or building materials.
Two wheeled wheelbarrows are more stable are often called garden carts. If the wheels are centrally positioned the load can be balanced and does not have to be supported by muscular force on the handles. They are much better for large and heavy loads - but require more space. Large gardens are often equipped with mini-tractors which can either drive a mower or pull a cart.
One-wheeled wheelbarrows are easier to maneuver around corners. Solid tyres are not liable to puncture but they make the barrow less manouverable. A barrow with a wide pneumatic tyre (eg 4"/100 mm) is lighter to push and has increased stability. As with bicycles, keeping the tyres well-inflated both resists punctures and improves the mechanical performance.