Nonhuman Races
Cavalry is traditionally a human practice, since typically only humans breed horses and are large enough to ride them.
Dwarves
Dwarves are not especially comfortable in the saddle and traditionally their armies favour infantry: mines, mountain passes and tunnels cannot readily be defended with cavalry. Fortunately, as most dwarves live in the hills and mountains, they rarely have to withstand cavalry charges as such manoeuvres require flat, open terrain. However, it's a big multiverse and if cavalry dwarves exist, chances are they are skirmishers using traditional dwarven resilience for lengthy harassing campaigns.
The main exception to this dwarven dislike for riding is found among the duergar, who breed 'steeders': fast-moving subterranean spiders. They have invented special saddles that allow riders to stay mounted even while the steeders climb the walls and ceilings of tunnels.
Elves
Out of the standard humanoid races, only the elves have the patience to breed horses and organise (term used advisedly) cavalry units. Their skill with magic and willingness to incorporate it into daily life results in horses of outstanding temperament and intelligence. Although the range of traits possessed by such beasts is beyond the scope of this article, simply giving horses the celestial template is a good place to start. The different elven subraces also make use of non-conventional steeds: grey elves occasionally have elite cavalry mounted griffons, hippogriffs or giant eagles; while aquatic elves may ride hippocampi. Wood and wild elves prefer to use arboreal creatures as mounts: for this purpose some may befriend dire wolves.
Gnomes
Horses are too large for gnomes to ride, and their preference for hills, forests and caves pretty much keeps them from using cavalry as shock troops. They can ride ponies and riding dogs without too much difficulty, but tend to use them solely for scouting and rapid skirmishing.
Halflings
Like gnomes, halflings are too small to ride horses effectively, but their natural agility puts them at an advantage when riding other animals, such as ponies or riding dogs. Since most other races are bigger than them, halflings know that to form heavy cavalry is an exercise in futility, except against races of comparable size such as goblins and kobolds. They use their ponies for a variety of other purposes: transport overland, mounted archery, scouting and skirmishing; although because they live on the plains they do not neglect charging tactics either, since their home terrain favours such actions.
Other Races
Orcish cavalry are pretty much unheard of: while orcs can easily grasp the idea that sitting on the back of an animal can reduce the amount of time spent walking, riding animals are usually seen as too bothersome to rear and keep. A horse in orcish care usually ends up as dinner. Some orcs, usually shamans, are exceptions to this rule and may ride a wyvern if they can capture one. Half-orcs may also be more inclined to ride horses, especially if they serve as members of adventuring parties.
Goblins, on the other hand, seem to have a natural aptitude for riding. They are infamous for forging alliances with packs of worgs; and a worg-mounted warband is a sight to be feared. They typically harass their foe's flanks, trying to force a retreat and picking off stragglers.
Centaurs make awesome cavalrymen: given that their bodies are composites of equine and human, they are able to combine speed and dexterity with the unique advantage that they cannot be dismounted. They can use a lance just as effectively as a mounted soldier, and more terrifyingly, can wield longbows without a problem. Luckily, they do not tend to have expansionist ambitions and prefer to live in the forest. This keeps their tribes from organising into overly large herds that are versatile enough to combine cavalry and archery with devastating results.