Every time my parents made cornbread they used an 8 x 8 Pyrex baking dish for it, and it seems most pictures of cornbread online it is made in a similar dish. I don’t have a small Pyrex dish but I do have a deep pan for making loaves of bread. Will anything horrendous happen if I make cornbread in this sort of pan?
You’ll get more loaf and less crusty goodness, but that’s about it. Do you have a cast iron skillet? Use that instead. Pre-heat it well and pour in your batter. Gets a good crust that way.
It would depend on if you completely filled the loaf pan. If so, then you would have a high, loaf shaped pan of cornbread and the texture may suffer. You would also need to adjust the baking time and/or temperature to compensate for the much thicker loaf.
I second the suggestion to use a cast iron skillet instead. But if you haven’t a cast iron skillet (and really you should, IMO ;)) then you could use the loaf pan, but fill it only an inch or two (and adjust baking time) or use a cake pan, or something with similar depth, almost any properly sized, oven-safe pan should work in a pinch; but you really can’t beat corn bread made in the skillet. 
Another vote for cast iron skillet. A greased and pre-heated 9" will work wonderfully
I don’t have a cast iron skillet. I currently have a non-stick skillet, but I didn’t like how the cornbread turned out. I’m in college and still moving every year or two, so I can’t fully equip a kitchen without giving myself a lot of headaches down the road. I also can’t financially justify buying new cookware I don’t absolutely need right now.
Were it me, a cast iron skillet would definitely be an absolute need. But it wasn’t always so I do understand where your coming from.
However, just in case you are interested, you can usually pick up a cast iron skillet at a thrift shop or garage sale for next to nothing- especially if you find one that was not well-tended. A big, heavy, rusty skillet can often be found for a dollar or less (around here anyway). Take it home, scrub it down to bare, shiny metal then season it up again and it will last another lifetime (or several!).
I use simple aluminium foil to make corn bread and it works fine. You can use any shape you want. As others have said, the amount of crust you want determains what kind of shape pan you should use to suit your own taste
Blasphemy! 
Another vote for finding a cast iron skillet or cornstick pan. If you have a muffin tin, your cornbread batter will also make some tasty corn muffins.
No, nothing hideous will happen. It may take a little longer to bake through in the middle, and the texture might not be exactly the same. But not necessarily worse. An alternative if you have a 9x13 Pyrex pan is to make a double batch. It will freeze fine if that’s too much to eat now. Muffins, if you go that route, will take a little less time to bake through.
You can buy a new, pre-seasoned Lodge 10-1/4" cast iron skillet for under $14 from Amazon. Used ones often pop up at thrift stores and yard sales for less.
You can use it for making cornbread, for pan-frying a nice steak, for bacon, potatoes, eggs… They’re very versatile, so they’re cost-effective. There is a learning curve, but a 10-1/4" cast-iron skillet is a very useful item to have.
EDIT: And upon re-read, I see MitzeKatze made the same point an hour ago. :smack:
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I usually make muffins with cornbread batter. Then I have nice, crusty goodness muffin tops. 
Or, make half a batch in the loaf pan.
Use what you have, but I too cast a vote for a cast iron skillet. Whatever you use you can test for when it is done like you would a cake. Stick a toothpick in the middle and see if it pulls out clean.
Actually, if the OP uses a loaf pan, using a knife to test would likely work better. Stentor, if you decide on the loaf pan, please post as to your success. I’m curious as to the results.
I am not Stentor but I was making cornbread anyway this afternoon so I mixed up a little extra to experiment with. The loaf pan(s) actually didn’t do too badly. One I filled up only about 2 inches (not precise, just eyeballing what a “normal” height for cornbread would be) and the other I filled nearly to the top (as I would for a quick bread loaf). Then I made a pan in the cast iron skillet as I would normally do. Here are my results:
The partially-filled loaf pan came out in texture and taste like any other cornbread made in pyrex or a cake pan. I did not adjust the temperature and the baking time was reduced a bit. I started checking for doneness at about 10-15 minutes, and it didn’t take much more than that. It was good but there isn’t very much of it. My 18 year old son ate it as one piece. The full loaf pan took longer, possibly 30-40 minutes, but I did not time it precisely because I was doing other things and just going by the clock, not using a timer. It came out more “cakey” and the bottom crust was just this side of burning. It was missing the nice crispy top that I can get with the cast iron and seemed moister than my cornbread is typically. Very tasty though and it sliced nicely. My girls slathered it with butter and devoured it while it was still hot. The cast iron skillet was my “control group” it came out like it always does, not too dry, not too moist and crispy where it should be. All of them tasted good.
So my conclusion is (as above) use what you have and enjoy your cornbread. Just be mindful of cooking time (and temperature) and be sure to start testing early if you are not sure. You should be able to turn out a decent, edible loaf of cornbread without too much extra effort no matter what pan you use. 
I fancy making some cornbread and (unusually, as it’s not a common ingredient here) I have both white and yellow cornmeal in the cupboard. Can someone who knows about these things recommend a recipe? I have a well-seasoned cast iron skillet I could use.
Cornbread recipes are going to be dependent on what you like and which “regional variation” you prefer. It can vary from savory to very sweet and almost like a cake.
I can toss you several very good and easy recipes that are equally good using yellow or white cornmeal, but you may not like them depending on your tastes. There is also (depending on brand and where you are) usually a basic cornbread recipe on the bag or box of cornmeal if you are buying it like that.
So…are you looking for a cornbread cake or something that you can use as an accompaniment for a big pot of beans?
Sorry, I know it is bad form to follow-up my own post, but here is a basic cornbread recipe:
Ingredients:
* 2 cups cornmeal
* 3/4 cup all-purpose flour
* 3 teaspoons baking powder
* 1 teaspoon salt
* 1 tablespoon sugar, or as desired (use up to 2/3 cup if you want sweet, cake-like cornbread but do not omit sugar thinking you don't want it sweet; a little sugar is necessary even for savory cornbread)
* 1 1/2 cups milk
* 2 large eggs
* 5 tablespoons butter, melted (you can use shortening or oil instead or for extra tastiness use bacon fat!)
Preparation:
Grease your skillet and stick it in the oven while heating to 400°. (200° C)
In a large bowl, combine the cornmeal, flour, baking powder, salt, and sugar.
In another bowl, whisk together the milk, eggs, and fat. Combine the two mixtures and stir until blended. (Do not overmix- just until blended, don’t try for a perfectly smooth batter or your texture will suffer.) Spoon into the prepared pan. Bake for 25 minutes, or until a toothpick comes out clean when inserted in the center.
What you see above is my basic cornbread, to a “T.” This is great as is, but can be customized as much as you want. I usually toss in a handful of grated cheese and a few diced pickled jalapeno rings. Maybe some crumbled bacon. But you have to cook it in cast iron for that really crispy crust.
I don’t know - both probably. Cornbread is such an alien thing here in the UK - so I suppose I’d like to try it in a variety of different ways.