Mutual indwelling of the divine Persons
At the risk of putting things in physical terms, perichoresis means that “all three persons occupy the same divine ‘space.'” In other words, we cannot see God without seeing all three persons at the same time. The three persons coinhere in such a way that the persons are always and forever with and in one another, yet without merging, blending, or confusion. Only by affirming the mutual indwelling of each other, can we worship our Triune God as truly three and truly one.
The reason it is important to grasp something of the divine perichoresis is that it protects us from a few errors when we think of the Trinity. One is called Arianism, another is called Sabellianism, and another is called Tri-theism. Arianism believes God the Father to be God alone, whilst the Spirit and the Son are lesser gods. Sabellianism (modalism) believes that God is one person that exists in different modes at different points in time (i.e. Father, Son and Spirit - but never all at once). Tri-theism believes that the Father, the Son and the Spirit are three separate gods that simply participate in an impersonal common divine essence (like different humans participate in human nature).
Since we have three co-equal persons who equally possess the divine essence, we steer clear of Arianism. Since all three persons are co-eternal, we steer clear of Sabellianism. And since all three persons co-inhere so as to exist in an inseparable manner with a single will and consciousness, we steer clear of Tri-theism.