If I have a stone what characteristics or tests can I use to determine what kind of stone it is.
There are no real tests you can do to determine the kind of stone, although you can get a practical appreciation for how it works — fine cutting or not, produces a scratchy or smooth or polished surface, how fast it cuts. Cutting speed is not always directly tied to how finely a stone cuts, although this may seem paradoxical the size and hardness of the abrasive particles can be the same in two stones but their spacing and how strongly they're bound together can vary greatly, q.v. the fast cutting of Japanese waterstones compared to natural American or European stones of approximately the same grit.
Anyway, stone IDs are a really tricky area and it may require an experienced collector to definitively identify a stone for you. But red stones aren't common so just as a quick guess it's just possible you're seeing an oil-soaked India stone, which would have originally been an orangey colour. Indias were one of the first synthetic hones, offered by Norton since the turn of the last century. They're made from aluminium oxide bound in a ceramic-like matrix.
If you want to get expert help in IDing your stones you can investigate this option by posting pictures on various knifemaking or straight-razor forums where members are usually very willing to help ID unknown stones for new members. Many have turned into stone collectors and have a vast knowledge of stones old and new.
You may need to clean the stone, as the current appearance of many old stones may often, in fact, completely mask their true appearance. In extreme cases a stone that is dark grey, greenish or brown turns out to be beige or nearly white after cleaning.
Here's roughly the same patch of one of mine before and after cleaning and lapping!
I am aware there are oil stones and waterstones. Are those the only two really. Besides diamond (which is relatively easy to identify) those are the only ones I see advertised.
Although many natural and synthetic Western stones are sold as oilstones note that they don't have to be used with oil.
Other than the fact that traditionally things other than oil were used for 'oilstones' almost all can be used effectively with water, a soapy mixture and sometimes even dry*.
If you are using oils broadly speaking this isn't always only one liquid, it can be anything from mineral oil at the viscous end to something finer like a light machine oil or honing oil (baby oil too if you don't mind the scent!), it was and still is quite common to see kerosene used, as well as that mineral spirits or WD-40 are often recommended.
Note: in general the finer the stone the better it responds to a lighter (less viscous) honing fluid, so it's worth trying various things to find out what works best.
I recall reading about Arkansas stones but that might be a brand name.
Arkansas stones should refer to a particular type of stone that actually does come from the state.
Unfortunately these days many kind-of-like Arkansas stones are sold under the name because it's a selling point. There are further designations of Arkansas stones into classes (e.g. Soft Arkansas, Hard Arkansas, white translucent, black translucent) but these too are abused to add value to stones that aren't what they look like.
Ditto the above with Washita, Charnley Forest, coticule and Turkey stones.
Cleaning the stones
As mentioned above if you post pictures of stones for an ID you may be asked to clean them first. But you should do this anyway, as old oil, dust, rust and metal particles will be removed from the stone and it will cut the way it should. The difference in abrasiveness between a gunked up old stone and one just cleaned can be remarkable.
You can clean stones in various ways, including at the very basic level just scrubbing down with soapy water and a scrub brush or nylon abrasive pad.
To get a stone really clean most of the time you have to resort to soaking. These days many will soak in Simple Green which I've not tried myself but apparently it works really well. It's more common to soak in a solvent of some kind, including mineral spirits but traditionally kerosene or gasoline were usually used because they were cheap and plentiful.
In cases of extreme oil buildup, especially when the oil is old and oxidised, caustic oven cleaners are sprayed on to break down the oil. As extreme as this sounds generally it does no harm to stones, whether natural or synthetic and it is very effective.
If you want a stone really clean it may be necessary to clean it in stages. You soak and clean off the surface, then let it sit for a while (usually weeks) giving time for more oil that soaked into the interior to leach out, which a further soaking can then remove.
*If using a stone dry it is vitally important to clean the surface regularly (ideally each time) to prevent loading or glazing, where the pores fill with metal particles and the item to be sharpened can just skate over the surface.