The bat pony race have several features to distinguish themselves from the other pony races (unicorns, pegasi, earth). Canonically, the bat ponies from the show are known to have grey coats, bat-like wings, and slitted-pupil eyes.
While only some of these traits are show canon, most of them are globally accepted headcanon.
Perhaps the first thing to be noticed on a bat pony are its wings, likely being the source of the race's namesake. While the feathered wings of pegasi usually share the same colour as their coats, bat pony wings tend to be on the more purple-hued side, or at least a closer colour to the pony's mane.
Some agree that bat ponies have a harder time at learning how to fly, mainly because of the fact that bird wings have few joints while bat wings have dozens. However, while flight may come easier to pegasi, bat ponies may have potential to be more efficient flyers.
Unlike their vibrant, pastel-coloured cousins, the coats of bat ponies tend to stay on the more grey side, varying in hues. There are several exceptions, but bat ponies seem to have a standard colour scheme.
That being said, their manes usually stay colourful.
They were never shown in the show, but fangs seemed to naturally tag along with the appearance of the bat ponies. It was perhaps the fact that, culturally, people tend to associate bats with blood drinking, despite only a few species of bat being known to drink blood. Yet, despite this, people put them on bat ponies anyways, including the ones considered fruit eaters. This trait gained canon support in the season 4 episode "Bats!" where Fluttershy was accidentally transformed into a vampire fruit bat pony, part of her transformation including a pair of fangs (though she used them to suck juices out of apples instead of draining blood from ponies or other animals). However, due to the fact that Fluttershy's vampire form was not actually a bat pony, this canon support is debatable.
Another distinguishable feature of a bat pony are their pupils, being thin and slitted as opposed to the wide, circular pupils of normal ponies. Despite not being a trait of real life bats, they were on Luna's guards, associating them as a nocturnal species.
Bat ponies tend to have fluffy ears. It is unsure where this trait comes from, but it seems to suggest that bat ponies are a little fluffier than normal.
Largely up for debate, many have thrown and bounced ideas around for the behavioural habits of the bat ponies.
Possibly the largest debate that still continues in the discussion of bat ponies is their diet.
The first source of food that comes to mind is usually blood or meat, as people tend to think of the bat ponies' fangs. Within the first threads on /mlp/, most seemed to accept the idea that bat ponies sustained through blood. It was perhaps this head canon that got the threads going.
However, some people found blood-drinking to be unsavory, and instead posed the idea of them being fruit bats. Usually, the people who accept this head canon believe that the bat ponies have the same diet as normal ponies, but have a particular liking to fruits.
Probably the most common headcanon,
Sometimes people like to mix-and-match, having both types of diets in their head canon, or combining them into omnivorous bat ponies.
Probably the most uncommon head canon regarding a bat pony's diet are bugs. Barring the fruit bats, bugs and insects are the main diet to bats in real life. However, people with this head canon are usually shot down quickly.
A strange trait added to the bat pony arsenal is the ability to use echolocation, usually in the form of screeching or at least the clicking of their tongue.
Some Anons suggested that the bat ponies have screeching contests to settle debates, or even to compete for loved ones. Some bat ponies may be better at it than others.
In almost every headcanon involving bat ponies, they are usually showcased as a nocturnal species. Having slitted-pupil eyes, most people implore that they are sensitive to light and can see well in the darkness. Coupled with echolocation, this helps provide a means for the bat ponies to navigate through the night. However, when exposed to bright environments, they may come across difficulties.
Several Anons in the bat threads promote the idea that the bat ponies have a special type of snickering. Being a race usually associated with the night, bat ponies are sometimes portrayed as being slightly mischievous, or at least playful.