Hi!

Yes, this is a MOSFET. In this CMOS digital logic application, they are indeed being used as switches. For an N channel FET, if the gate voltage is substantially higher than the source voltage, the FET will be "on" and allow current to flow from drain to source, and if the gate voltage is close to or below the source voltage, the FET will be off and no current can flow. P channel FETs work the opposite way (current flows freely from source to drain when the gate voltage is substantially lower than the source voltage).
You can see from your circuit then that if either A or B is near ground (low), one of the top two P channel FETs is on, which will allow current to flow from VCC into the output, setting the output to a logic high. If A and B are both near VCC (high), then both N channel FETs are on, and current flows from the output back into ground, pulling the output down to a logic low. This is indeed a NAND gate.
CMOS is so-name "complementary MOS logic" because it uses both N channel and P channel FETs. You can also make logic using only one channel by substituting a resistor for the other side. This resistor often lives outside the IC, and you end up with an "open drain" or "open source" output. Other logic topologies exist that use the more commonly known bipolar junction transistors (BJTs), instead. The most common is "TTL" such as the 74xx (no inner letters) series logic.
MOSFETs are actually bidirectional. For an N-channel MOSFET, as long as the gate voltage is sufficiently more positive than the source voltage, current can flow either way. However, conventional MOSFET construction techniques end up with a so-called "body drain diode" with the anode connected to the source (since the body is connected to the source) and the cathode connected to drain. This diode allows current to flow from the source to the drain whether or not the MOSFET is "on", albeit with a full diode voltage drop (typically ~0.7V). Sometimes this can get in the way, but it can also serve as a built-in freewheeling diode in inductive load switching applications, and you can buy MOSFETs with this built in diode highly characterized for this and similar purposes.
However a MOSFET does have a very usable "partial conduction" range: it's called the saturation region, unlike a "saturated" bipolar transistor which is "saturated" when VCE is as low as it can get, and the bipolar transistor is acting as a switch. Confusing, I know. The BJT equivalent of the MOSFET's saturation region of operation is the "active" or "forward active" region.
Many car stereos (used to) have MOSFET linear output stages, though "class D" amplifiers, which are similar in many respects to a switch mode power supply, are very common now due to their higher efficiency and minimal cost in terms of sound quality.
The "switch" region on a MOSFET is commonly called the "ohmic" region because it effectively acts like a low value resistor (milliohms to dozens of ohms, typically). A BJT being used as a switch acts like a constant voltage drop (typically about 0.2V).
You can often interchange a MOSFET and a BJT with just a few circuit tweaks. MOSFETs are typically used in switch applications due to their effectively zero static gate current requirements and lower on state losses at low-medium currents.
A hybrid of the two also exists: the IGBT. These have a voltage controlled gate like a MOSFET, but the on-state characteristics more closely match a bipolar transistor. These are often used in high power applications. There also exist other kinds of FETs, though they see limited action these days. JFETs are still around, mostly as inputs on op-amps.
And nope, basic electronics haven't changed a ton in 40 or so years (the FET is about that old), but how they are used definitely has. Miniaturization has a lot to do with that. Your quad core processors have some 800 million plus transistors on them. That was unthinkable in the 70s.