Forging is a process where hot metal of the rough shape is then very forcibly squished by exact shape dies, severely compressing the metal or alloy molecules. There are internal tensions created within the structure, which ultimately resist sheer and tension stresses by having a reserve of counterbalanced "forces" due to the myriad of tiny forge-created stress nodes inside. Dies are not necessary for forging, as any blacksmith or swordsmith is creating similar stresses when beating hot metal with a hammer as it cools. However, if the part (or sword) is reheated again, much of the gain can be lost as the molecules relax their cramped positions.
There is also quenching, which is taking a metal part that is very near a eutectic (phase change) point and suddenly cooling it in an oil or water bath, perhaps even liquefied gas. This creates similar strength properties, but should not be considered forging as the gain is shallow and typically does not extend through the part. This typically followed by "annealing", which is a very controlled heating and slow cooling process which relieves SOME of the surface stresses that might lead to cracks or shattering. If the part is irregular in size (like a connecting rod with a big end and small end) the quenching process is also likely to cause warping which must be removed, typically mechanically by bending. I'm not aware of this being used much at all for engine parts.
Most mass-production of engine parts are done with casting, which is unlike forging in that molten metal is "simply" poured into a near-shape mold and allowed to cool. Molecules of the alloy are allowed to move around as they need to, and the part has little internal stress. This is a less-costly process than forging, as molds are much cheaper to make than forging dies--as well as many other reasons. Forged parts often start out as castings. In high-horsepower applications and other cases where RPM and/or cylinder pressures (BMEP) are very high, the added expense of forgings is well worth it.
BMW, Porsche, Corvette, Ferrari, etc. will use forgings where cost is less of a concern vs. performance and perception. Another cruical bonus, is that parts can be forged with less material for a lower weight, while still being stronger than an apples-to-oranges casting. This is HUGE in something like a connecting rod (plus a piston), where most failures are NOT due to compression from boost, but from tension stresses fatigue from changing direction after Top Dead Center. Each entire reciprocating assembly might only weigh a few pounds, but image the stress if you had to throw that away from you and instantly jerk it back to you -- 15000 times a minute, 250 times a SECOND on a literbike or modern F1 engine. THIS is why high compression, high RPM, high boost engines use forged internals... not so much because of the pressure, but because of the higher tension forces created by higher-weight castings -- forces that are (I think I recall) CUBED with respect to RPM, and squared with respect to weight. [that recollection might not be exact] ... Going any further in weight reduction suggests materials like titanium or exotic nanoparticle carbon metal composites yadayada, which are far beyond answering your question or my wallet.
ALL of these processes require exact machining afterwords to achieve the proper exact dimensions for bearing fitment and threads for assembly.
Metallurgy is incredibly interesting. 6000 years ago artisans were making swords that still have not been fully recreated to this day. Despite incredible gains in alloy and metal working technology, I still feel the only huge difference between the art of alchemy and the science of metallurgy, is that metallurgy actually works!
I'd love a set of forged Manly or Crower rods for my Saabaru EJ205 project, but $1000-$1400 is likely 4 or 5 times the cost of perfecly serviceable stock parts for a stock vehicle. I won't even mention the cost of a forged crankshaft. And then there are these piston things... [sigh]