Change of stress on words followed by enclitics – double accents on certain words
The enclitic particles are a small number of short words that combine (usually follow) closely with verbs or nouns (mostly) in order to signify possesion of something or action on something. These words are μου, σου, του, μας, σας, των, με, σε, τον, τους (some of these words might have homonyms that are used differently). Some examples:
Το παιδί μου (My child)
Τα αυγά μας (Our eggs)
Φίλα με (Kiss me)
As these words have a very close connection with the noun or verb they "belong" to, the sentence's stresses may change. There is a very good introduction here: http://www.slideshare.net/dimitrakapnia/enclitics
This phenomenon causes a change in the spelling of some words that reflects this. Words that have an accent on the antepenultimate (third from the end) syllable also take a second accent on the ultimate syllable when followed by an enclitic. Some examples:
Το άλογό σας (Your horse)
Τα γενέθλιά μου (My birthday)
Φίλησέ με (Kiss me; This is not a mistake, some verbs have double endings, this one is φιλώ or φιλάω)
As I already said, this is not only a matter of spelling, but also of word stress. You can try Google Translate's Greek text-to-speech to see the difference.
Many of the sentences in the course that should follow this rule don't. Apart from teaching a wrong spelling, this has the added effect that the audio files that were synthesized using those spellings sound a bit off.
PS. There are also cases (not in this course I think) where more enclitics are combined, and this can change the stresses and spelling even in case of words accented in the penultimate (second from the end) syllable. An example:
Δώσε μού το (Give it to me)
Here an accent is also put on the first enclitic.