term "fat" when referring to NMOS and PMOS
Microchip designer here...

I've never heard of a "fat" MOS transistor, but there is such a thing as a "thick oxide" MOSFET transistor, sometimes referred to as just "thick" transistors.

Typically, a MOSFET transistor is designed such that the electrical performance is good for a particular voltage range. For example, 0.7 to 1 volts. That means that it switches pretty fast, and its leakage current is acceptable.

But sometimes you want a transistor to be a little speedier switching. So you'll make the gate oxide a little thinner than typical. This is known as a "thin oxide" MOSFET transistor.

A thin oxide transistor's Vt (threshold voltage) is lower than the typical gate oxide transistor's Vt, so it will switch faster and stronger. But it also means there's more leakage, which means there's wasted power and a build-up of heat. If it wasn't for that, we'd just design all the transistors with thin-oxide.

As for thick-oxide transistors, that's the opposite of a thin-oxide transistor. The Vt (threshold voltage) goes up. Switching speed goes down. Leakage current and heat also goes down. The reason thick-oxide transistors are used is mostly because the voltage applied to these devices is much higher than typical transistors. With higher voltages, you need more of an insulator (oxide layer) at the gate. Otherwise it will stress the gate terminal too much and cause catastrophic damage to it.

Why do you need to handle different voltages on the same chip? The answer is that in the "core" of your design, where you're actually handling all of the logic functionality, you will use whatever voltage is optimal for that particular semiconductor process technology and your design goals. But when you have to interface your chip with the outside world, you'll have to use whatever voltage is best to do that. Typically, that voltage is much higher than the core voltage. So the core might be at 1V, but the I/O interface might be at 1.7 to 3V. And for those, you'll need to use thick-oxide devices.

If you're referring to the width of the transistor when you say "fat" MOSFETs (which I've never heard before), then yes it's just a wider transistor than normal. The thing is, there really aren't "wide" transistors and "thin" transistors. In almost every semiconductor process technology I've seen, the width of the MOSFET transistors varies along a range. So for example, from 10 nanometers to 10 micrometers in the same technology. That's a pretty big range. And so in any given microchip design, you'll see transistors with all different widths. It just depends on what kind of functions you're trying to perform on the chip. Rarely are you going to see transistors that are super wide (like 10 microns wide when most transistors are under 0.1 microns). But I suppose if you do see that, then those you could conceivably call "fat" transistors.