The spider's position relative to the rest of the speaker - highlighted in red.
The spider's only two purposes are to ensure that the diaphragm will only move linearly, keeping the voice coil centred in its place, and to make sure that the voice coil never rubs the permanent magnet, as such could muffle and impede the speaker's playback, or could even prevent the speaker from working altogether.
The reason for this component's odd name is that before they had a resemblance to spider's legs, however most speaker spiders today are simply disc shaped with ripples, and a larger amount of imagination is required to see a spider.
Most spiders are usually made from fabric and is stuffed with a resin which stiffens it, and spiders must be manufactured extremely carefully to ensure it will only move on one axis.