As I am sure you have noticed, Hungarian is a language that uses additional endings to convey meaning, rather than additional words.

For example:

kalapjaiméit those of my hats, accusative
kalap + jai + m + é + i + t
hat + plural possession + 1st person + that of hat + plural + object
hat ... his hats ... my hats ... that of my hats ... those of my hats .... (object)

When a word, or a word with an ending, ends in a vowel, the vowel may be lengthened when it has an (additional) ending added to it.

The vowels which display this most are a and e.

apa father
-m my -
apa + m = apám my father

körte pear
-t (object)
körte + t = körtét pear (object)

As you will see in the nouns section, different cases will often require a vowel between a consonant-final stem and an ending. The usual endings a -o- for back-vowel words and -e- or -ö- for short and long front-vowel words respectively. But low vowel words will always take -a- (back) or -e- (front). Such words are usually flagged as such in the dictionary.

A good example is ház, which would normally (according the the regular rules) take -o- to make házom, házot, etc, but in fact it takes -a- to make házam, házat.

A fleeting vowel is one that gets squeezed out when endings are applied. An examle is eper. Normally the accusative would make this eperet, but the second e is ejected toi make epret. This is a fleeting vowel.

This is a subtle different but it's very important. It can totally change the meaning of the sentence.

Compare:
én láttam a házat. I saw the house.
to:
én láttam a házát. I saw his house.

The formation of the former is
ház + at = házat. (the) house (object).

The formation of the latter is
ház + a = háza. his house.
háza + t = házát. his house (object).