When should I use the verb "work" over "working"?

What is the difference between:

I am working today

and

I work today

What is the right form?

Let's take them in reverse order:

I work today.

This means that you are going to work today. It is not said if you are currently working. Examples would be:

I can't go to the beach. I work today.

Aww, the boss screwed my day off. I work today, but at least I have tomorrow off.

Now, this is still not what most native speakers would say. They would probably say "I have work today," or "I'm working today." Which brings us to your first example:

I am working today.

This will mostly be used when you mean that you are going to work today. It can also be used if someone asks you what you're doing while you are at work (or on a lunch break). It would not normally be heard at work, because it should be obvious to any questioner who is present that you are working (and if it's not obvious that you are working, any question about what you are doing would be intended as a rebuke). The most common use case would be if you got a call on your cell from a friend:

Caller: Hey, what are you up to? I thought we could grab some beer and hit the beach.

You: I can't. I'm working today. Can I get a raincheck?

EDIT

You may also get some use out of looking at this question and answer, which I helped craft (along with some of the EL&U moderators and other bright lights on this site). It's not perfect, and it's still a work in progress, but we designed it to give non-native speakers a visual introduction to what the statements of time mean in English.

The first sentence combines a form of to be (am) with a present participle (working) to form the continuous tense. The sentence "I am working today" tells me that either 1) you are working right at this moment while you speak to me or 2) you are going to be working all of today.

The second sentence is the simple present tense...though a bit unusual. The simple present tense is usually used to tell us something general. Unlike the present continuous tense described above, the present simple doesn't often have anything to say about time. So if I were to say simply "I work," it would mean that I generally work, without any specific mention of when, or how long. But by putting the "today" at the end, you are making a declaration of time which is odd for the present simple.

So, to the answer: I would, in general, use the first construction. Though an exact answer would be dependent on the context of the situation.