In my heavily homebrewed version of Greyhawk I reworked most of the non-human species. I wanted them to behave and feel similar to their traditional versions but come up with some actual biological and cultural reasons for their behaviour rather than using the overly simplistic D&D alignments (which I don't use at all).

Elves' emotions change very slowly and need long, sustained experiences to really trigger them. But when triggered they also last much longer, sending them into weeks or months or years long emotional states (love, sorrow, rage, etc). This means that fleeting events have almost no emotional impact on them, which makes them seem cold, aloof and in some cases cruel by human standards (e.g. single moments of violence inflicted to them or by them register no emotional response). Their culture has extremely intricate customs of etiquette and behaviour to manage a society of what are essentially sociopaths in their day-to-day lives. Elven armies have corps of elves that are in the midst of long-term rage (often triggered by the impacts of the war) that are essentially in a months long berzerker state.

Orcs are no less intelligent than humans but have very poor long term memories, so they aren't good at tasks that require extended study and building up of knowledge. But this means they live very much in the moment giving them powerful insight and wisdom, their perception and thoughts on what is currently happening aren't influenced by biases and expectations. They have very high pain tolerance and casual violence is an inherent part of any discussion or debate among them, they express their thoughts and feelings in inherently physical ways. They breed very quickly which leads to regular expansion and conflict amongst themselves and as they expand into new areas.

And there's a whole thing with goblins that after I'd put it together someone pointed out to me that it is almost literally the xenomorph aliens from the Alien franchise (goblin queens giving birth to huge litters, goblin births picking up genetic material from what the queen has eaten so if she eats humans they get hobgoblins, etc). They have a hyper pack mentality, where the bigger the group they are in the more energetic and stupid they are. So a single goblin creeping around will be terrified of everything, but a big swarm of goblins will be suicidaly reckless. They also have photographic memories for navigating their huge cave and tunnel systems, which makes them good at learning and copying stuff (though they aren't good at inventing new stuff). The pack mentality lessens over time as they age, so leaders are usually older goblins that can think more clearly, but also a lot of older goblins leave the hive to make their own way in the world.