Davide's Notes

Nice page at http://www.andrews.edu/SEM/semtech/HebrewTutorial/lesson_03.htm. Here are the basic rules for the definite article, summarized:

1. Remember that the gutturals are א ה ח ע ; we need to consider also ר as a "special" consonant in this context
2. With words not beginning with a guttural or ר, the article is usually written הַ + dagesh, as in הַמֶּלֶך
   
3. With words beginning with one of the gutturals ה ח, and with words beginning with יְ מְ (note the shewa), the article remains הַ but without the dagesh (implicit), as in הַחֶרֶב
4. With words beginning with one of the gutturals א ע, plus ר, the article becomes הָ, without the dagesh, as in הָאִישׁ
5. With words beginning with an unaccented הָ or עָ, and always with words beginning with חָ, the article becomes הֶ, as in הֶהָרִים

In other words, the article is:

הַ : with all non gutturals, and with gutturals ה ח (in this last case, and with יְ מְ , there is just no explicit dagesh)
הָ : with gutturals א ע and ר (no explicit dagesh)
הֶ : with חָ and unaccented הָ עָ (in practice, pay attention to when a qamets is present with a guttural)

There are exceptions: some words change pointing when the article is prefixed to them, for example
אֶרֶץ becomes הָאָרֶץ

Now, remember that with prepositions the pointing becomes very important, because what will happen with prepositions directly joined to a word having the definite article is that we'll have the consonant from the preposition, and the vowel from the definite article. So, for example:

מֶלֶך king
הַמֶּלֶך the king
לַמֶּלֶך to the king, using the preposition לְ, to