Ha esik az eső, ne hagyd otthon az esernyődet!

Ha means if.

In this sentence, it introduces the condition: Ha esik az eső... = If it rains / If it’s raining...

It very often comes at the beginning of the clause, especially in straightforward sentences like this one. That is the most natural neutral order.

Esik az eső is a very common Hungarian weather expression. Literally, it is something like the rain is falling.

Hungarian often uses this kind of structure for weather:

• esik az eső = it is raining
• literally: the rain falls

You can also hear just esik when the context is clear, but esik az eső is a very normal full expression.

Because eső is treated as a normal noun phrase here, and Hungarian commonly uses the definite article in this weather expression.

So:

• az eső = the rain
• esik az eső = literally the rain is falling

This does not necessarily mean one specific rain in the way English the often does. It is just the standard Hungarian way of saying it.

Because this is a negative command.

In Hungarian:

• nem negates statements
• ne is used with commands, wishes, and similar forms

So:

• Nem hagyod otthon az esernyődet. = You are not leaving your umbrella at home.
  This is a statement.
• Ne hagyd otthon az esernyődet! = Don’t leave your umbrella at home!
  This is a command.

Because the sentence has a definite direct object: az esernyődet.

Hungarian verbs often change depending on whether the object is:

• indefinite
• definite

Here the object is definite, so Hungarian uses the definite conjugation.

Compare:

• Hagyj itt valamit! = Leave something here!
  indefinite object
• Hagyd itt a könyvet! = Leave the book here!
  definite object

So in your sentence:

• hagyd is the correct 2nd person singular imperative with a definite object

Yes. It is addressed to one person informally: te.

You can tell from:

• hagyd = informal singular command
• esernyődet = your umbrella with the informal singular possessive ending

A formal singular version would be:

• Ha esik az eső, ne hagyja otthon az esernyőjét!

Otthon means at home or home in this context.

Grammatically, it is an adverb, not a noun with a case ending here. It tells you the place where something is left.

So:

• hagyd otthon = leave it at home

This is a very common combination in Hungarian.

It contains several pieces:

• esernyő = umbrella
• -d = your, for te
• -t = accusative marker, showing it is the direct object

So the overall meaning is your umbrella as the object of the verb.

The form is not just a simple mechanical stacking of letters, because Hungarian often uses a linking vowel in possessed object forms. That is why you get esernyődet, not a shorter-looking form.

A useful related form is:

• esernyőd = your umbrella
• esernyődet = your umbrella, as the object

Because in Hungarian, a possessed noun can still take an article.

So Hungarian can say:

• az esernyődet = your umbrella
• a könyvemet = my book
• a kabátját = his/her coat

This is normal Hungarian. English does not usually say the your umbrella, but Hungarian structure works differently.

Not completely. Hungarian word order is flexible, but different orders give different emphasis.

The version in your sentence is a natural neutral order:

• Ne hagyd otthon az esernyődet!

If you want to emphasize the umbrella, you could say:

• Az esernyődet ne hagyd otthon!

That sounds more like:

• Your umbrella, don’t leave that at home.

So the original sentence is a very natural everyday version, not the only possible one.

Because Ha esik az eső is an introductory subordinate clause.

Hungarian normally separates this kind of if-clause from the main clause with a comma:

• Ha esik az eső, ne hagyd otthon az esernyődet!

This is similar to English punctuation in:

• If it rains, don’t leave your umbrella at home.

Yes. Hungarian present tense often covers both ideas, depending on context.

So Ha esik az eső can match English:

• If it rains
• If it is raining

The exact English translation depends on the situation, but the Hungarian form itself is completely natural for either kind of everyday warning or advice.