It’s a good question, and, even if my first langage is french, I needed to think about it.

Le jour du contrat, tu dois préparer un chèque pour le loyer du premier mois et la caution.

In this case, you can designate the contract, and the month for rent.

La caution, c’est en général deux mois de loyer.

Here, like the rent is for 2 months, you can’t designate the rent, and it’s not one month.

I don’t know if you can understand the difference, it’s difficult to explain this in english

It took me a moment, but I see it. It is because of how “rent” is being used in the second sentence: “months of rent”.

Because it is being used to describe the guarantee, rather than be the subject of the sentence itself, its article (the le) is dropped. The subject of the second sentence is “la caution” which is described as “deux mois de loyer”.

Le jour du contrat, tu dois préparer un chèque pour le loyer du premier mois et la caution.
La caution, c’est en général deux mois de loyer.

On the day of contract(on the contract day) … the rent of the first month.
The guarantee, it is generally 2 months rent worth.

It depends whether you’re talking about something specific and singular, or something more general and non-specific: "du" designates something specific of which there is only one, while "de" designates something that isn’t singular.

• "Le jour du contrat" (we are talking about one specific contract)
• "Le loyer du premier mois" (there is only one specific first month)
• "deux mois de loyer" (rent is something general, there isn’t only one specific rent)

For example:

• "une chaussure du danseur" (a shoe from the dancer: we already know which specific dancer we’re talking about)
• "une chaussure de danseur" (a dancer’s shoe: a shoe that would be suited for dancing)
• "une balle de tennis" (a tennis ball)
• "une balle du garçon" (a ball that belongs to the boy)
• "un morceau de pain" (a piece of bread)
• "un morceau du pain" (a piece of the bread loaf)
• "un morceau de piano" (a piece of music played on a piano)
• "un morceau du piano" (a piece of the piano – indicating the piano is broken in pieces)