Your sentences would be as follow correct:
- (Én) eszek (valamit) = I eat something. – indefinite conjugation
- (Én) eszem a gulyást. = I eat the "gulyás". – definite conjugation
Every Hungarian verbs can have 2 conjugation-forms (excluding the intransitive verbs, for example jönni = to come, menni = to go):
1. indefinite conjugation (= alanyi ragozás):
this one shows the number and the person of the subjective
Én eszek = I am eating. – who does eat? only this is important. I can’t say with these sentence exactly,what I eat (valamit = something, an apple = egy almát, but I did’t say, which one)
2. definite conjugation ( = tárgyas ragozás)
in this case we have a definite object. It means, I know exactly, what I speak about.
Én eszem a gulyást – what do I eat? the „gulyás” – I know exactly, which „gulyás” I eat
Be careful! Én egy gulyást eszek. = I am eating a „gulyás”.– in this case I didn’t say, which one I eat. It is simple a „gulyás”. (indefinite conjugation)
It is not official but we learned it in the school as follow:
- indefinite conjugation ( = alanyi ragozás) – if you can say „valamit” with the verb (something)
/ I see something. = (Én) Látok valamit./
- definite conjugation (= tárgyas ragozás) – if you can say „azt” with the verb (that in accusative)
/I see that . (the dog / the door) = (Én) Látom azt. (a kutyát, az ajtót)/
Read more here (Point 5):
http://en.wikipedia.org/wiki/Hungarian_verbs#Definite_and_indefinite_conjugations