Why CMOS?
First I would explain why CMOS exists; Oli[n] - :) - have said the advantages, but let me take a step back.
The need for complementary gates is due to the fact that the simpler gate concept is based on the concept of pull-up and pull-down; this means that there is a device (a transistor or a set of transistors) which pulls the ouput high and another device to pull it down.
The enhancement nMOS, which is the best performing MOSFET, needs a \$V_{GS}>V_T>0.7V\$ in order to work; for this reason, it works well as a pull down device, but not so well as a pull up. Hence the idea to use the pMOS, that is a bit worse (because holes move slower than electrons, but this is another story) but acts perfectly as a pull up.
So complementary because you use two device that behave in the opposite way and are thus complementary. Then, the logic is inverting because nMOS (that pulls down) requires a high voltage to switch on and pMOS requires a low voltage.
But why MOS is good?
And some additional informations: as also Olin said, the main reason for the spreading of MOSFET technology is that it is a planar device, that means that is suitable to be made on the surface of a semiconductor.
This is because, as you can see in the picture, building a MOSFET (this is a n-channel, the p-channel in the same substrate requires an additional doped region called n-well) basically consists in doping the two n+ regions and deposing the gate and the contacts (very very simplified).

BJT transistors now are made also in MOS-like technology, so printed in a surface, but basically they consist in three layers of semiconductor differently doped, so they are primarily meant for discrete technology. In fact, the way they are now built is creating these three layers at different deepness in the silicon, and (just to give an idea), in recent technology they occupy an area in the squared micrometer order or so, while MOS transistors can be built in <20 nm technology, with an overall area that can be in the order of about 100 nm². (picture in the right)

So you can see that, added to the other properties, MOSFET transistor is buch better suited (in thoday's technology) to achieve Very Large Scale Integration, or VLSI.
Anyway, bipolar transistor are still widely used in analog electronics, for their better linearity properties. As a side note, a BJT is faster than a MOSFET built with the same technology (meant as transistor dimensions).
CMOS vs MOS
Note that CMOS is not equivalent to MOS: since the C is for 'Complementary', it's a particular (even if widely used) configuration for MOS gates, while high speed circuits often use dynamic logic, that aims basically to reduce input capacitance of gates. In fact, trying to push the technology to the limit, having two gate capacitances (as CMOS have) at the input is a source of loss of performance. You could say that it's sufficient to increase the current delivered by the previous stage but, to make an example, 2x charging speed requires 2x charging current, that means 2x conductivity, which is achieved with 2x channel width, and - surprise - that doubles the capacity.
Also other topologies, like pass-transistor logic, can simplify the structure of certain gates.
About interfaces
Changing topic, when talking about microcontrollers and interfaces, it's important to remember that the high input impedance of CMOS gates makes very important to ensure that Input/Output pins are never left floating (if they have protection, this is ensured internally), as their gate can be exposed to external noise and assume unpredictable values. So stating that a device has CMOS characteristics should also advise you of this.
(It's a long answer, probably full of typos: feel free to edit it to correct it)