The outside layer of hair, the cuticle, is a bunch of cells layered on top of each other like scales. You can feel this by pulling you hand down a few strands (which feels smooth) and then back up the strands the “wrong” direction (which feels slightly sticky or rough).
The shampoo used to lift dirt and oils off the hair also “opens up” the “scales” of the cuticle, making them stick out… From what I read a while ago, this has something to do with the pH balance of soap… When the cuticle is open like this, the hair feels sticky or rough, and it tangles with itself very easily (think velcro). Conditioners are designed to flatten the cuticle back down so that the hairs have a “smooth” direction once again, and can slide against each other without getting caught in each other. (Again, I think it has something to do with the pH, but I don’t totally understand chemistry.)
Conditioners that smooth the cuticle out best tend to have phrases like “manageable hair/improves manageability” or “softens hair” or “detangles.” Sometimes “smooth and/or sleek and/or silky” indicates that as well… Although a smooth cuticle will give hair a shiny appearance, conditioners that advertise making hair “shiny” or having a “________ appearance” aren’t necessarily smoothing out the cuticle. (Though they may offer the hair protection via silicones or oils that give the hair a shiny appearance.)