Bombilla is the straw used when drinking mate from a gourd. The word is Spanish, and can also mean light bulb. It also literally means small pump. The origin is the Latin word bulbus, which means onion.
You can understand this when you see the object. A long tube that ends in a small onion-shaped strainer. It can be unscrewed and opened in two parts for cleaning. The safest tip to avoid clogging your bombilla is not to mess around with it.
Sure, you can drink mate without bombilla, but if you want the right feel, it's simply part of the culture.