perfect tense

I have some questions about the perfct tense...

I have broken my leg.

Is it the same thing to say the following as above?

I broke my leg and it is still broken.

What is the difference between these two sentences?
*Did you already eat?
*Have you already eaten?

Thank you in advance.

[RonBee]

"I have broken my leg" and "I broke my leg and it is still broken" are not the same thing.

I will attempt to give a more complete answer later.

[Casiopea]

The statement "I have broken my leg" needs more context. It could mean,

"I have broken my leg before"

which could mean one of two things:

1) it is still broken or
2) it is no longer broken.

Without some sort of added modification, we don't know what "I have broken my leg" is in reference to, so we can't say for sure whether the leg is still broken or not. Try:

"I have just now broken my leg" means, it is still broken.

The same holds true for the Simple Tenses:

"I broke my leg."

We don't know if the leg is still broken or not. We need more context. Let's add modification:

"I broke my leg 10 years ago." (The leg is not broken)
"I broke my leg an hour ago." (The leg is broken)

=====

In terms of how 1. and 2. differ, well, it depends on whom you're talking to. Some people say 1. and 2. mean the same thing, whereas other people say they mean two different things:

Did you eat? (Simple Past = direct reference to time / informal)
Have you eaten? (Present Perfect = indirect reference to time / formal)

If you're writing an exam, refer to 'have eaten' as time without boundaries, and 'ate' as time with boundaries.

In many cultures of the World, it's often considered impolite to ask people in authority direct questions. In Japanese, for example, speakers end questions with words like "...don't you think?" which allows the askee a choice in their response. In English, the use of Have...? works in a similar way. The reason being, time is not an issue with the Present Perfect. It expresses "some unknown time in the past". It's that 'unknown' part of the Present perfect that speakers draw on when asking "Have you...? The assumption is that it allows the askee a choice in not having to give the time s/he had eaten. The same holds true for the Simple Past: the speaker doesn't have to say when s/he ate. But, it's different because it implies a time, whereas the Present Perfect does not imply a time.

Did you eat? (implies a specific time)
Have you eaten? (does not imply a specific time).

:D

[RonBee]

I would like to add one small thing to Cas's wonderfully complete explanation.

"Did you eat?" is not normally be used by itself, while "Have you eaten?" often is. Examples:

• Did you eat breakfast?
  Did you eat lunch?
  Did you eat your food?

:D