What hair conditioner basically does is replace the natural oils that shampoo strips away when you wash your hair. They usually contain one or more fats, oils, or synthetic compounds that behave like fats and oils, surfactants, and a bunch of other stuff that makes it more marketable(perfumes, dyes, preservatives, etc.).

The fats or oils (natural or synthetic) leave a smooth, waxy coating on the hair, making the hair smooth and shiny. Hair is made up of an outer layer called the cuticle, a second layer called the cortex, and an inner layer called the medulla. Hair tangles when the lifted or frayed edges of the cuticle layer on one hair get caught on the cuticle layer on another hair. The way conditioning fights tangles is that it strengthens the cuticle or, where cuticle cells have broken away, forms a protective layer over the cortex. This coating essentially smooths out the rough edges along the cuticle layer and thus can keep your hair from tangling as easily. The coating also seals in moisture and helps reduce static electricity, both of which help in avoiding bad hair days... ;-)
Hope that answers your question!

I think that most hair conditioners work by sticking to and coating your hair. That's why hair is shinier when conditioner is used. Hair tangles less and is easier to comb because of the coating too--the conditioner coating acts like a lubricant, like oil on a bunch of spaghetti.

But does it truly do anything to your hair? No. Hair is ~97% protein, but it is essentially dead tissue. That means that describing your hair's "health" doesn't make any sense. What conditioner might actually do is make you like the look or feel of your hair better, like the way polishing a table might make it look or feel nicer. If you blow-dry your hair, the coating might protect your hair a little bit from drying out, becoming rough, and getting split ends.

Companies that make conditioners use statements about your hair's "health" to try to trick health-conscious people to buy their products. Just think of furniture polish: it might make your table top look nicer, it might even protect it from a little wear and tear, but the wood in the table is dead, so it can't be made any healthier.

Thanks for the excellent question! Scientific questions like yours can affect our everyday lives and buying habits, too!