Expanding a bit on four pillars (who knows his stuff)

It's not dissimilar from wine aging. Hops, being very volitile, will fade in prominence as time goes on. I'm high abv beers, the alcohol "heat" can also fade, as has happened in a 19% abv beer I brewed. Oxidation can also cause various other things to happen, like enhancing Sherry flavors or Port flavors.

It really depends on the beer and what your looking for. Stouts and barleywine are the most common things to age because they're big and complex and have room, so to speak, to change as they age.

This is also ignoring things like lambics, which some Brewers age themselves and mix things like 3 year old beers with fresh beers before packaging. A place like Cantillon really straddles the line between beer and wine.