Effect > Amplify permanently changes the digital sound package and the blue waves reflect those changes. If you constantly changed the volume of the sound that way, up and down and up and down, you will eventually cause some minor sound damage.
The slider to the left of the track is a “fake” volume effect and is not burned into the music until you export (or I think, Render). The down side is you cannot see the blue waves change, and you could forget you changed the show that way.
Also, the Mixer toolbar has no effect on exported sounds - it only affects the playback level.
The Track Gain Slider affect the “Track”, so they also affect the exported file when you finish the project and they affect the “mix” if you Mix and Render tracks.
PCM may use integer or floating point data.
Audacity supports tracks that are 16 bit integer. 24 bit integer and 32 bit float (floating point). 32 bit float is strongly recommended for audio processing.
32 bit integer PCM has 2^32 (4,294,967,296) possible values that are evenly spread out between +/- 1.0
32 bit float spreads out its bit values differently so that they provide greater precision for small values (close to silence) and can also represent values greater than +/- 1.0 (thus 32 bit float format does not clip at 0 dB, which can be a life saver when processing audio).
32 bit float format is primarily used as a production format (because of it’s extreme precision). 32 bit float is way over the top for playback purposes as even the best professional hardware has no more than 23 bits precision. The most compatible audio format is 16 bit 44100 Hz sample rate (CD quality). For extreme audiophile quality you could use 24 bit at either 44100 Hz or 48000 Hz (whichever is the native sample rate for your hardware - usually 44100 Hz for audio only and 48000 Hz for video).