Are there any simple rules for choosing the definite vs. indefinite (vs. none) article?

There isn't a simple rule, as the definite and the indefinite article are used in different cases.

• Eat an apple before going to the cinema.

  The sentence is not referring to a particular apple; it can be a random one.

• Follow the President.

  The sentence is not referring to a random president.

The difference is not between talking of something the other person knows, or not.
In the second example, the person I am talking to could know all the presidents present in the meeting, but he will understand to which president I am referring.

• Albert taught himself to play the violin.

• Worry about the future.

  In this case, the article is used to make a generalized reference to something rather than identifying a particular instance.

• They placed the African elephant on their endangered list.

  African elephant is referring to the whole species, not to a single elephant.

• She is a McFry.

  The sentence is referring to a member of the family McFry.

More broadly, an interesting colloquial usage of "the" that I've seen is related to subject familiarity. "Learn the computer once and for all", belies a lack of familiarity with computing in general by lumping hardware, software, networking, work, leisure, etc., into an arms-length abstraction "the computer". Not that there's anything wrong with this. Being totally unfamiliar with sports, I might say "the baseball" (referring to the game), because I am essentially warning my listener not to expect any depth from me on that one.

I'm not entirely sure what mechanism is at play, but whether it is used whimsically or unconsciously, it seems to be a concise and reliable construct.

Yep. I hate articles too. BTW I am wondering why nobody mentioned that "countable/uncountable/plural" issues? As I understand:

• You cannot use a/an with plural forms
• You cannot use a/an with uncountable stuff (say, milk, sugar, water)
• Some words are kind of "dual" and can represent both countable and uncountable stuff. Example: icecream

Correct me if I am wrong.

I've actually put some time into thinking about this and I think the most basic use of "the" and "a/an" has to do with what the speaker/writer assumes about what the listener/reader already knows. A common explanation is that "the" is for definite references, but "a/an" is used for indefinite. Though this may sometimes be the case, it isn't always.

Ex. "I bought a car today." is most certainly not just any car. It's the exact car that the speaker bought. The speaker is saying, "I bought a specific car, but I assume you don't know which one I'm talking about."

Ex. "I bought the car today." which is saying, "I bought a car and I assume you already know which one I'm taking about." (Probably because the speaker has mentioned it to the listener before.)

So if the speaker is wrong in their assumption that the listener already knows, the listener would normally respond, "What car?" In other words, "Please tell me which car you are referring to, because I don't know."

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[The text below is in response to Araucaria's comment. -- Thanks, btw.]

The zero article is the plural for "a/an", but I think it points to a category.

Ex. "I like fast cars." = "I like things that fit into the category of fast car."

Ex. "I like cake." = "I like food that fits into the category of cake."

The plural for "a/an" = "speaker assumes listener/reader doesn't know" is "some".

Ex. "I bought some pretzels on the way home" (assumes listener doesn't know which "pretzels")

The plural for "the" in this case is "these/those". Ex. "I bought those pretzels on the way home" (assumes listener does know which pretzels are being referred to) Note also that if the speaker's assumption is wrong, the listener will usually respond, "Which pretzels?"

There are some other functions of "the", "a/a", etc. as well, but I can't remember what they are at the moment.

I think the "the computer" situation is a case where "the" refers to a general category, as in "The dog is a domesticated animal" or "The dodo bird is extinct." I suppose you could say, "The computer is one of mankind's most influential inventions."

If you were to say, "Learn about the baseball.", it would probably sound to most people like you're talking about an actual ball, rather than the sport. It would clearer to say, "Learn about the game of baseball." which I'm pretty sure would also mean baseball as a general category (of game).

I can’t for the life of me figure out where to use a and where to use the — and where there is no article at all. Is there a simple rule of thumb to memorize?

The standard rule you always hear:

“If a person knows which item you are talking about then use "the"

. . . doesn’t clear things up for me, as I have no idea whether or not they know.

That's fine. All you need to do is tweak this guideline (it is not a rule) to

“If you think/assume a person knows which item you are talking about then use "the" (or some other definite determiner)

See the second sentence in the Wikipedia article on English articles, which is cast similarly to mine (Use of the definite article implies that the speaker assumes the listener knows the identity of the noun's referent (because it is obvious, because it is common knowledge, or because it was mentioned in the same sentence or an earlier sentence)).

In all the following, you assume your listener knows which item (referent) you are talking about, by

(a) context
(b) shared knowledge between you and your listener
(c) knowledge of the world in general.

Examples:
You have just seen a movie with a friend. You say to him:
--Wow, the movie was great wasn't it. a and b

You make lunch arrangements with a friend. You say to him:
--I'll meet you at the cafeteria at 1pm. b (you eat at the same cafeteria each day, or a there is only one cafeteria at which it is possible to eat)

You tell your friend where you left his mail. You say:
--I left your mail under the doormat to the door of your house. This could also be b but c can come into play: most houses have doors, so using the door relies on this knowledge; you could further specify the front door if you felt the need. As for doormat, well that is b, if you both know that the door has a doormat. Or it could be a/c if your friend's wife just bought a doormat and placed it there, your friend can easily figure out which doormat you are referring to.

Your friend wants to know where your wife is. You say:
--She is out walking the dog. Mostly b

There are some idiomatic uses that nevertheless can be explained by this guideline: You have a landline and actully three phones connected to it, so that all three phones ring at once when someone calls. You ask your friend to see who is calling. You say:
--Could you answer the phone? Even though three phones are ringing, no matter which phone you pick up, it will be the phone. It is similar with other things around the house:
--Could you answer the door?

However, there are plenty of idiomatic uses that this guideline does not cover. If your friend asks what your wife is doing and she is reading something, if it is any newspaper, you just say:
--She is reading the newspaper. This is true whether or not you assume your friend can identify which newspaper it is, because the newspaper is idiomatic. It is different from book or magazine, for example, in which you would not use the unless you assume your friend knows which one it is.

On the other hand, a person can use the indefinite article even though he knows you can identify which referent he is talking about:

I. Joe buys a hotdog on the way home. He gets home and he is holding it where you can see it. Joe can say
--Hi. I bought a hotdog on the way home. (Here it seems that the fact that it is 'any old' hotdog trumps the fact that you can easily identify which one it is.)

II. A more subtle case: Joe and his wife Mary are riding home. Joe asks Mary what she wants to do when they get home. Mary can say:

I want to read a book when I get home.

Now, under many analyses, this could be taken to mean 'any old' book, or a book that Joe cannot identify. But in actuality, Mary can say a book even though she knows that Joe knows which one she is talking about. (Say it is the one she reads two pages a night from, each and every night.) For whatever reason, Mary makes an indefinite reference to it, even though she knows Joe knows which one. Why does she do this? Because she can. (English grammar allows this.) And it shows that what I gave at first is only a guideline, not a hard and fast rule.

The opposite is at play too. A person can use the definite article even when he knows his listener cannot identfy which referent he is talking about. Joe can say to Mary:
--Man, the experience I had at work today is not one I would ever like to happen again.
As an opening line in his spoken text, Joe knows that Mary cannot identify which or what experience he is talking about. At this moment this seems to break the guideline I mentioned. But it demonstrates another function of the and that is to bring up a new subject of conversation. Despite many references that will say that a speaker will use a/an to bring up a new topic, we actually use the for this also. And Joe will go on to identify what experience he is talking about. So actually this falls under a context, because the context will make the referent clear as Joe continues his report of what happened at work.

But, yeah as for a simple rule, the guideline I suggest works better than the one you quoted.

For completeness, since this is the canonical question on articles, there is the case of adjectives, which of course are not nouns and don't take an article at all.

• He is Italian.
• He is an Italian.

The word Italian looks the same in those instances, but the first is an adjective describing the man and functioning in the same way as "He is hungry"; the second is a noun, a native of Italy. Nouns can take articles, and the rules and usages are described in other answers. Adjectives don't take articles.

The simple rules you ask for:

• If there is only one of it in the world, use "the", as in the Statue of Liberty;

• When it is a thing or place the listener knows, use "the", as in the shed in our backyard;

• If it is a place of general purpose, use "the", as in the college or the hospital (sometimes, we do not use an article); and

• Otherwise, if you have brought it up before, use "the", as in getting back to the other story.

The rules for "a" are, essentially, the converse:

• If there are many of it, use "a", as in a strange man;

• When the listener likely does not know it, use "a", as in I went on a rollercoaster once;

• Otherwise, if this is your first time talking about this, use "a", as in a friend of mine.

Simply, if the noun is abstract, as in money, fame, or beauty, use no article. However, these, too, take articles in some contexts: we say Money does not bring happiness but Do you have the money?, Her name and fame spread worldwide but The fame got to him, Beauty is in the eyes of the beholder but (we may say poetically) Hers is a beauty unparalleled.

Note that a new subject matter is introduced with "a" but takes the definite "the" afterwards: I saw a movie last night. The movie was about a girls-only all-American baseball team that kept folks entertained during WWII.