आज जन्मदिन है।

• आज = today
• जन्मदिन = birthday
• है = is

A very literal translation is Today birthday is. Natural English rearranges that to Today is the birthday or, more naturally in context, Today is [someone's] birthday.

Hindi usually puts the verb at or near the end of the sentence. So instead of English Today is birthday, Hindi gives आज जन्मदिन है = Today birthday is.

That is a very normal Hindi pattern:

• time expression first: आज
• noun/predicate: जन्मदिन
• verb at the end: है

Not by itself, at least not as clearly as English does.

If you specifically want Today is my birthday, the full and most natural sentence is:

• आज मेरा जन्मदिन है।

The shorter आज जन्मदिन है। can work if the context already makes it obvious whose birthday you mean, but on its own it feels incomplete to many learners and often to native speakers too.

Hindi does not need a dummy subject like English it in this kind of sentence.

English often uses it just because the sentence needs a subject. Hindi does not do that here. So Hindi can simply say:

• आज मेरा जन्मदिन है।

There is no separate word for that English it.

है is the present-tense form of to be: is/am depending on the sentence.

Here it links जन्मदिन with the rest of the sentence, so it functions like is in English:

• आज जन्मदिन है = Today is birthday
• more naturally: Today is a birthday / Today is [someone's] birthday

In ordinary spoken Hindi, है is important here. Leaving it out would sound incomplete, except in very casual, poetic, or headline-style language.

Because जन्मदिन is singular.

• है is used with singular subjects
• हैं is used with plural subjects, and also with respectful you in some sentence types

Here the subject idea is birthday, which is one thing, so है is correct:

• मेरा जन्मदिन है = My birthday is...

You would not use हैं here just because the birthday belongs to someone important. The noun जन्मदिन is still singular.

Yes, जन्मदिन is normally treated as one compound noun meaning birthday.

It is made from:

• जन्म = birth
• दिन = day

So it is literally birth-day.

A careful pronunciation is roughly:

• janm-din

In everyday speech, many speakers say something closer to:

• janamdin

So if you hear either kind of pronunciation, that is normal.

Yes. In everyday Hindi, many people say:

• आज मेरा बर्थडे है।

That is very common in casual speech.

The difference is mostly register:

• जन्मदिन = more standard / more Hindi
• बर्थडे = very common in modern casual speech

Both are understood.

That mark is called the danda. It works like a full stop or period.

So:

• आज जन्मदिन है।

means the same kind of sentence ending as:

• Aaj janmadin hai.

In modern Hindi, you may also sometimes see a regular English-style period, especially online, but । is the traditional Devanagari sentence-ending mark.

Yes. Hindi word order is flexible, especially with time words like आज.

For example:

• आज मेरा जन्मदिन है। = neutral and very common
• मेरा जन्मदिन आज है। = also natural, with a little more focus on today

So आज does not have to stay only at the beginning. The meaning stays basically the same, but the emphasis can shift a little.