Learn Hardcore French: Ce mois est difficile pour ma famille. - This month is difficult for my family.

Ce mois est difficile…
Neutral “this month,” with context usually making clear whether it’s the current month or a specific month you’re talking about (often “this month we’re in”).
Ce mois-ci est difficile…
Very clearly “this month (here, the current one)” and contrasts strongly with other months. You’d use it if you really want to stress “this particular month (as opposed to others).”
Ce mois-là était difficile…
“That month” (over there / back then), referring to a month already mentioned or located in the past story.

So ce mois is fine and natural when the context already makes it clear which month you mean (often the current one).

Why is there no article, like le mois? Why not Le mois est difficile pour ma famille?

French uses ce (this/that) instead of le (the) when you want to point to a specific thing.

Le mois est difficile…
“The month is difficult…” sounds odd without context, as if there is only one month in existence. You’d normally follow it with something like Le mois de janvier est difficile… (“The month of January is difficult…”).
Ce mois est difficile…
“This month is difficult…” clearly identifies which month you mean in the situation (usually the current one).

So here ce is the natural choice because you’re talking about a specific month in time, not “the month” in general.

Why is it ce mois and not cet mois? How do ce / cet / cette / ces work?

Ce / cet / cette / ces are all forms of “this/that”:

ce before masculine singular nouns starting with a consonant:
ce mois, ce livre, ce père
cet before masculine singular nouns starting with a vowel or silent h:
cet ami, cet homme, cet hiver
cette before feminine singular nouns:
cette semaine, cette maison
ces for any plural nouns:
ces mois, ces familles, ces amis

Mois is masculine and starts with a consonant sound /m/, so you must use ce mois, not cet mois.

Why is difficile after est and not before mois, like in English “difficult month”?

In French you have two common patterns:

Noun + être + adjective
- Ce mois est difficile. = This month is difficult.
  This describes a state or quality.
Adjective + noun or noun + adjective inside a noun phrase
- un mois difficile = a difficult month

In your sentence, you’re using a full clause with the verb être:
Ce mois (subject) + est (verb) + difficile (adjective).
That’s why the adjective appears after est, not before mois.

If you wanted a noun phrase, you could say:
C’est un mois difficile pour ma famille. = “It’s been a difficult month for my family.”

Does difficile agree with mois? Why doesn’t it change form?

Yes, difficile is agreeing with mois, but the masculine singular form of difficile looks the same as the feminine singular one.

Masculine singular: un mois difficile
Feminine singular: une semaine difficile
Masculine plural: des mois difficiles
Feminine plural: des semaines difficiles

In your sentence, mois is masculine singular, so difficile keeps the basic form (no extra s or e visible).

Why is it ma famille and not mon famille? And is famille singular or plural?

Famille is grammatically feminine singular in French: la famille.
Possessive adjectives agree with the gender and number of the noun, not with the owner.

So:

ma famille = my family (one family, feminine)
ta famille = your family
sa famille = his/her family

Even though a family contains many people, grammatically it’s one unit (singular), so the verb and adjectives that refer to it are singular too:
Ma famille est grande. — “My family is big.”

Could I say pour la famille instead of pour ma famille? What’s the difference?

Yes, but it changes the meaning:

pour ma famille = for my family (specifically the speaker’s family)
pour la famille = for the family (maybe a particular family everyone knows about, or “family” as an institution, depending on context)

In most personal contexts, if you mean the difficulties your own relatives are experiencing, you say ma famille.

Is pour the only preposition I can use here? Could I say à ma famille?

In this meaning, no: you should stick with pour.

difficile pour ma famille = “difficult for my family” (they are the ones having a hard time)
difficile à ma famille is not idiomatic French here and sounds wrong.

With adjectives about how something affects someone, French very often uses pour:

C’est dur pour lui. — It’s hard for him.
Cette période est difficile pour nous. — This period is difficult for us.

What is the difference between Ce mois est difficile pour ma famille and C’est un mois difficile pour ma famille?

Both are correct and close in meaning, but the focus is slightly different:

Ce mois est difficile pour ma famille.
Literally: “This month is difficult for my family.”
You’re pointing directly at this specific month and describing it.
C’est un mois difficile pour ma famille.
Literally: “It is a difficult month for my family.”
This sounds a bit more like a general comment on the type of month you’re going through. Very natural in conversation, often used to summarize:
Cette année est compliquée. C’est un mois difficile pour ma famille.

Both can usually be used interchangeably in everyday speech.

How do you pronounce mois and est here? Are there liaisons?

mois is pronounced roughly like [mwa] (the s is silent).
est is pronounced [ɛ] (like “eh”).

In Ce mois est difficile:

No liaison between ce and mois:
/sə mwa/ (not /sə.z‿mwa/)
Between mois and est, you can make a light linking of the vowel sounds, but you don’t add a consonant:
/mwa ɛ difisil/ → said smoothly, almost like /mwaɛ difisil/

There is no pronounced ‘s’ or ‘t’ linking mois and est in standard speech.