A wheelie bin is a fairly large bin (usually around 200 litres capacity). They are typically around 120cm tall, and have a cross section of 50cm square, and are used for residential rubbish and recycling.
They have a hinged lid to keep the rubbish and smell in. Due to their size, they are too heavy to lift like a conventional "trash can" style bin, so they sit on two wheels at the back. This allows them to be wheeled around like a trolley.
Garbage trucks with an robotic arm are used to pick up the bin, turn them upside down to empty them, then put them back down, reducing the amount of labour required to collect the garbage.
Since waste management is a local government issue, the style of bin and collection frequency varies widely. Some areas have switched to wheelie bins to allow less frequent collections, but some areas haven't because they feel that a large bin does little to discourage waste production.