The name handcart is often used to refer to two-wheeled carts that are pushed or pulled by hand. In the olden days, it was widely used in a variety of trades, such as delivering parcels, selling goods and hauling materials.
Want to sell something at the local market? Throw them in that 2 x 6 feet cart and find a spot. Need to deliver bread around the neighborhood? Lay them carefully over the handcart and make your rounds. Records indicate that handcarts were employed all over the world, with references to it in literature dating back to the second millennium BC.
Handcarts differentiate themselves from carts and wagons by both form and function. The latter were bigger, were hauled by animals (such as horses and cows), and frequently ran on three or more wheels. They were also used to deliver unusually hefty loads. Handcarts, on the other hand, often rolled on two tires and were meant to be handled by people.
Starting from the mid-1930s, handcarts have been increasingly replaced by other implements. Small-package deliveries can now be accomplished much more conveniently by vans, tricycles and other similar vehicles. Nobody lugs around a cart-full of stuff as a mobile store anymore (save, perhaps, for a few far-flung places).
Nowadays, handcarts remain in use, although the avenues that employ them have been severely cut down. Most of the time, we see handcarts relegated to dirty jobs - such as those in construction sites, farms and gardens, where heavy stuff will need to be carried within immediate distances. We also see it in some places where equipment need to be hauled around, such as airports (for luggage), harbors and hotels.
Handcarts today can be called by many names, most of them to suggest its intended use. In airports, they're often called luggage carts or airport trolleys; in hotels, they can be referred to as baggage carts; in construction, many call them hand trucks (hey, a "truck" does sound more fitting for a building site than a "cart" after all).
Whatever name you wish to call them - hand trolleys, tool carts and probably dozens more - they all do the same thing. They carry your heavy loads, roll on wheels and move with the aid of your strength - either through pushing or tugging.
Many places still sell handcarts today. If you're looking for one, try hitting the home improvement and gardening stores. They're also readily available online, mostly from sites that specialize in construction equipment and specialty supplies (e.g. for hotels, airports and such).