A hologram is a three-dimensional image captured within a two dimensional plane. Similarly, the holographic principle states that the amount of information within an N dimensional space is proportional to the area of that space's N-1 dimensional enclosing surface and not to the volume of the space. It is as if the information were projected holographically onto that surface. At present, the holographic principle is a conjecture.
The argument for the holographic principle relies on quantum gravity. Heisenberg's uncertainty relation between energy and time is ΔE•Δt=h/4π. This means that, to know when something is happening to some minimum accuracy (Δt), you must use an energy of at least ΔE to pin it down. Now E=mc2 and the "m", which is mass, distorts space/time geometry with its gravity and when that gravity is powerful enough, it changes the space/time geometry in a way that isolates "m" from the rest of the universe - a black hole. So, if you measure shorter and shorter units of time, you'll eventually run up against a limit - the Planck time (=~10-43 seconds) - when your test particle becomes a black hole. The Planck distance is the distance light could travel during the Planck time (=~10-33 cm) and is the shortest measurable distance.
Now imagine one bit of information - either present (=1) or absent(=0). It's minimum diameter is the Planck distance and its area is on the order of the Planck distance squared. It turns out that if you add a second bit, the minimum enclosing volume is two times this "Planck area" and so on. Thus, the amount of information that can be contained within a given space is the number of bits times the Planck area - a two-dimensional quantity even though the information is contained in three dimensions. It is as if all of what you can know about the universe, is what's projected on a two-dimensional screen through which you view or experience that universe.
See also: black hole entropy, information theory