Philosophers during the modern era wrestled with various philosophical problems. Some of these were cast as new problems, although many had been vexing philosophers since the start of the profession. One of these is the problem of the external world: how do I know that my perceptions match what is real? Another is the mind-body problem: how does the mind interact causally with the body? Leibniz attempted to address these problems (and more) with his monads.
For Leibniz, monads are the fundamental entities that make up the world. They are immaterial and, although they have qualities, they have no parts. There are supposed to be an infinite number of these entities and, apparently, they all perceive. On most interpretations, each monad is a mind. However, the monads do vary in their degree of mental capabilities and they range from the most minimally perceiving monad to the supreme monad (not to be confused with the supreme Dalek or a nacho supreme) which is, of course, God. The higher sorts of monads are conscious and aware while the lower sorts presumably are not. As such, while your soap perceives (think about that the next time you lather up) it is not conscious (which is probably best for both of you).
While all these myriad monads perceive, this perception is not (as Leibniz sees it) a perception caused by external objects. As Leibniz famously claimed, the monads do not have windows and (in addition to making it hard to enjoy warm spring days) nothing enters or departs from them. However, each monad is supposed to mirror all of reality. While I usually use the analogy of a bowl full of polished ball bearings as an analogy to illustrate that bit, the analogy rather obviously fails badly. But, I do think it is a nice image.
While this windowlessness might seem rather odd, it does enable Leibniz to solve two problems with one nad, monad, that is. First, the mind body problem is elegantly solved: reality is fundamentally mental (which I am sure you have long suspected) and hence there are not two distinct metaphysical types to have relationship problems. There is but one type and, perhaps even better, there are no causal relations between these monads (well, aside from God's act of creation, but God is always mucking up things). Thus, these problems are solved. Well, sort of anyway. Second, the problem of the external world is also solved. Monads do not perceive what is outside of them, for there are no windows via which they interact with an external world. The split between experience and reality that allows the problem of the external world to gain traction simply is not there, hence its wheels spin futilely. Or would, if problems had wheels.
Assuming that you buy this, there are still some obvious problems remaining. One is the matter of addressing the intuitively plausible view that we are perceiving the same reality and that we seem to interact. For example, as I type the blog my husky (a husky monad) is watching. I believe that she is perceiving me doing this and I believe that I am perceiving her perceiving me and that she is no doubt wishing that I was handing her some treats rather than typing. So, how does this work with monads?
For Leibniz the answer is very straightforward. In the beginning, God created all the monads and placed "in" each one all its experiences (sort of like downloading a whole movie before starting to play it). Being really amazing, God makes sure that all the monads are in sync (no, not in the boy band). So, back to the husky example, when I have the experience of seeing my husky and she has the experience of seeing me, we are not "really" seeing each other. Rather Isis (my husky) is having an experience in her mind as if she were seeing me and likewise for me. While I do suspect that husky hair could actually get into a monad, there is no actual causal interaction between us. However, the experiences are in a state of pre-established harmony and hence it all works out. Really.
Not surprisingly, this has caused some people to wonder why this does not just collapse into solipsism. After all, if all my experiences are pre-loaded, then I should have them whether there are any other monads or not. By Occam's Razor, one might argue, it would seem simplest to hold that I and I alone exist. Or, at best it is just me and the creator-which sticks us (or rather just me) into the problem raised by Descartes. Perhaps even worse, if the God monad perceives everything perfectly, then it would seem to entail that everything is just a quality of God's mind. This is, of course, pantheism and something the sane generally endeavor to avoid whenever possible. As such, let us quietly close that door and sneak away.
Now that all those problems have been successfully ignored, there is the obvious problem of space. If we are just immaterial monads, then the space we perceive would thus clearly not be space in the usual sense of a box in which God keeps his stuff. Also, what we take to be extended (three dimensional) objects cannot actually be three dimensional in the usual sense.
Leibniz solves the first problem by taking space to be a system of relationships between what a monad experiences. To use a contemporary example, think about "moving" around in 3D video game like Halo or World of Warcraft. It seems like you are moving through space because of the relationship between the elements of your experience, yet there really are not three dimensions in the usual sense. Space is merely a matter of perception and relative to the experiences.
In regards to objects appearing to be extended, this is also a matter of perception. While Leibniz uses the analogy of a rainbow, the video game analogy works even better. In video games we experience what seem to be extended objects, even though they are not actually extended. Rather, the extension is something of an illusion. Likewise for the monad's experience of extension-it is all in their minds. The monads look inwards and see all that can be seen.