The compression effect in music amounts to increasing the amount of energy/loudness in the sound within the limits of the playback device to avoid "clipping".
Put very simply, it increases the amplitude of the input sound. Its parameters are controlled by internal timers, external triggers, timers and signals ("side-chaining"). For example; lowering the bass ("ducking") during the critical milliseconds of the kick-drum so that the volume of both can stay maximised while their sum stays within maximum amplitude.
On voice-over, in radio and podcasts it makes voices sound even and "boomy/strong".
Originally probably necessary way to make recordings listenable in cars and noisy environments with lower spec speakers.
Nowadays producers use it to "sound louder" and thus make stronger impressions.
Lookhead limiting is more commonly used to sound louder, not compression. Compression is usually done for flavor. It’s not that great at making things louder, because traditional compression actually exaggerated the spikes in amplitude at the start of percussive sounds, which pop and dance music has a lot of, requiring additional work to tame. Modern drum processing usually uses a combination of compression (sometimes with upwards compression), lookahead limiting, and saturation.