У них хорошие отношения.

In this sentence, у них is a very common Russian pattern meaning something like with them / among them / in their case.

Literally, у often means by, near, at, but in many expressions it is used more abstractly. Here, у них хорошие отношения means that they have a good relationship or their relations are good.

This is a normal Russian way to express what English often says with have:

• У меня есть время = I have time
• У него друзья = He has friends
• У них хорошие отношения = They have a good relationship / They are on good terms

So у них does not mean physical location here. It introduces the people whose relationship is being described.

Because они is the nominative form, used for the subject of a sentence, while них is the form required after the preposition у.

Russian pronouns change form depending on their grammatical role:

• они = they
• их / них = forms used in other cases, including after some prepositions

After у, you need the genitive case, so:

• у меня
• у тебя
• у него
• у неё
• у нас
• у вас
• у них

Also notice that after a preposition, Russian often uses the form with н-:

• у них
• к нему
• с ней

So у они would be ungrammatical.

Yes — in a sense.

In the present tense, Russian usually does not use a word for is/are in sentences like this. So:

• У них хорошие отношения.
  literally looks like
• With them good relations.

But the understood meaning is:

• They have a good relationship or
• Their relationship is good or
• They are on good terms

This is completely normal in Russian. Compare:

• Он дома. = He is at home
• Она врач. = She is a doctor
• У них хорошие отношения. = They have a good relationship

In past or future, a form of быть appears:

• У них были хорошие отношения. = They had a good relationship
• У них будут хорошие отношения. = They will have a good relationship

Because хорошие has to agree with отношения in number, gender, and case.

The noun отношения is:

• plural
• nominative here

So the adjective must also be:

• plural
• nominative

That gives:

• хорошие отношения

Compare:

• хорошее отношение = a good attitude / a good relation (singular neuter)
• хорошие отношения = good relations / a good relationship (plural)

This is standard adjective-noun agreement in Russian.

In Russian, отношения is very commonly used in the plural to mean relations or relationship between people.

So even when English often says relationship in the singular, Russian often prefers the plural:

• хорошие отношения
• плохие отношения
• дружеские отношения

The singular отношение exists, but it often means something slightly different, such as:

• attitude
• approach
• relation in a more abstract sense

For example:

• Моё отношение к этому отрицательное. = My attitude toward this is negative.

So in your sentence, the plural отношения is the natural choice for talking about how two or more people get along.

Here отношения is in the nominative plural.

Why? Because it is the main noun being described:

• У них = prepositional phrase giving the context
• хорошие отношения = the core statement, literally good relations

You can think of the structure as:

• [With them] [good relations]

Since отношения is the thing being asserted about them, it appears in the nominative.

Yes, Russian word order is flexible, though some versions sound more natural in certain contexts.

The neutral order is:

• У них хорошие отношения.

You might also hear:

• Хорошие отношения у них.

That version can sound more contrastive or emphatic, as if focusing on good relations or contrasting them with someone else.

In ordinary conversation, У них хорошие отношения is the most natural default order.

Yes, that is also correct, and it means almost the same thing: They are on good terms.

Compare:

• У них хорошие отношения.
  Slightly more neutral; literally They have good relations.

• Они в хороших отношениях.
  More like They are in a good relationship / on good terms.

Both are common. The first is a very basic and useful pattern. The second uses:

• в
  • prepositional case
• хороших отношениях = in good relations / on good terms

So both are good to know, but the original sentence is extremely standard.

Yes, Между ними хорошие отношения is also possible.

That version emphasizes the relationship between them:

• между ними = between them

Compare the nuance:

• У них хорошие отношения. = They have a good relationship / They are on good terms
• Между ними хорошие отношения. = There are good relations between them

The difference is small, but между ними focuses more explicitly on the connection linking the two sides.

Них is pronounced roughly like neekh, with a soft n and a vowel similar to ee in see.

A few helpful notes:

• у них sounds like oo neekh
• the stress in них is on the only vowel: них
• in normal speech, у них flows together smoothly

The whole sentence is pronounced approximately:

• oo neekh kha-RO-shi-ye at-na-SHE-ni-ya

Stress:

• у НИХ
• хоРОшие
• отноШЕния

No. Отношения is broader than that.

Depending on context, it can mean:

• personal relations
• professional relations
• diplomatic relations
• family relations
• romantic relationship

So У них хорошие отношения could mean:

• they get along well
• they are on good terms
• they have a good working relationship
• they have a good personal relationship

If Russian wants to make the romantic meaning clearer, the context usually does that, or a more specific phrase may be used.

This pattern is very productive. You can replace the adjective to describe different kinds of relationships:

• У них плохие отношения. = They have a bad relationship.
• У них дружеские отношения. = They have friendly relations.
• У них деловые отношения. = They have a business relationship.
• У них напряжённые отношения. = They have tense relations.
• У них близкие отношения. = They have a close relationship.
• У них сложные отношения. = They have a complicated relationship.

This makes у + pronoun/person + adjective + отношения a very useful pattern to remember.