Burst mode, or continuous shooting mode, is a feature found on digital cameras which allows the photographer to shoot a series of photos one after the other.
The speed at which images can be taken is known as frames per second (fps), and it varies hugely depending on the price and model of camera. Top DSLR cameras can have speeds of up to 7 fps, and they can shoot several dozen RAW or JPEG shots at a time. The number of frames you can shoot is defined by the size of the buffer, where images are stored before they're processed.
Some DSLRs are specifically designed to shoot at a higher fps at the expense of resolution. These cameras are generally used for shooting fast-moving sports, where a series of images shot at high speed will often be necessary.
But even the lowest specification of cameras will have a small burst mode, which can be very useful for shooting anything that won't stay still for long ... like small children!