His old house: بيته القديم.
His house is old = (his house) is (old): Here, (his house) is subject, and (old) is predicative; Predicative means we telling information about (his house). Onun evi eski.
In Arabic, the predicative is mostly (if not always) indefinite, which is in this case قديم.
However, to say His old house (i.e. Onun eski evi), the adjective here is "attributive". It describes the noun (his house) only and it is not telling information about the house. In Arabic, attributive adjectives come attached to the noun and they must be like the noun in definition, number, gender. (His house) by nature is defined (even without AL الـ), because "His" acts like a definition (that is, you are talking about someone's specific house not a random house), and thus if we want to say (his old house) we must use القديم instead of قديم - the adjective here becomes defined like (his house). So, just by adding AL or removing AL, we can change the adjective (and the sentence) from predicative (without AL) to attributive (with AL) in this case.
Hope this clears up things :)