Why amps, pre-amps, integrated amps???

Separate preamps and power amps have a certain cache that dates back to the days when tube equipment was the only amplification option.

The stuff was big and heavy, especially so as an amp's wattage rating increased. A higher power integrated amp could quickly become too big and heavy to be practical as consumer gear.

There was also the issue of the high power components on the same chassis interfering with the small signals handled by the pre-amp (especially the phono section).

The ultimate expression was a separate preamp and two monoblock power amps.

In the 1950s and 60s, tube integrated amps were typically limited to lower power units. (The classic integrated Dynaco SCA-35 was 17 tube watts a channel.)

These days, especially with class-D amplification, weight and chassis size are no longer an issue for high power integrated amps. There are some who would argue that the improved signal isolation of a separate chassis for each section is still worth it, but even modern integrated gear is incredibly quiet.

However, if nothing else, the cache of "separates" remains a good selling point for high-end gear.

A receiver is a tuner (to get radio), a preamp (selects what input you want to listen to and controls volume) and an amp that drives the speakers. Basically 3 boxes in one box. If the receiver has a phono section and a headphone section then make that 5 boxes in one box.
An integrated is a preamp and an amp in one box, with phono and headphone depending on the model.
Preamps simply select the input and control volume. And some preamps include phono and / or headphone as well.
Amps just take the input from a preamp and amplify, drive the speakers.
Receivers, preamps, amps, headphone amps, phono preamps all exist because "different strokes for different folks".
Most people on this forum would say that the best sound would come from 5 separate boxes (if you were interested in all 5 functions). I run an integrated amp with a separate phono amp. I'm not interested in radio or headphones. I might get better sound from a separate preamp and amp, but it would also cost a lot more, I'd need another AC outlet, more shelf space, etc. so practical considerations also exist on why people choose what they have.
There are slight variations (example an amp can take a signal directly from a CD player that has a volume control, things like that) but for the most part I think I covered it pretty well.
Oh yeah, and a CD player is also a transport and DAC, so you can consider that 2 boxes in one box also.

I was led to believe that better performance from separates could be achieved with being able to better split up and manage the power supplies. Not sure if that is true or not. In a general sense, it is always better to have more options so the more you separate components, the more options you have to mix and match and upgrade, etc. To me, I like the integrated approach although I prefer to have a stand alone phono stage that offers more flexibility than you will usually find in an integrated with a phono. An integrated also means you have less to worry about in terms of interconnects and power cords.