Мне нужны две штуки.

Because Russian uses a different structure for to need.

The pattern is:

• кому? + нужен / нужна / нужно / нужны + что?

Literally, Мне нужны две штуки is closer to:

• To me are-needed two items

So мне is the dative case of я, meaning to me.

Compare:

• Мне нужен чай. — I need tea.
• Мне нужна машина. — I need a car.
• Мне нужны деньги. — I need money.

So even though English says I need..., Russian says something more like To me, ... is needed.

Because нужны agrees with the thing that is needed, and here that thing is две штуки, which is treated as plural.

The adjective-like word нужен changes depending on gender and number:

• нужен — masculine singular
• нужна — feminine singular
• нужно — neuter singular
• нужны — plural

Examples:

• Мне нужен билет. — I need a ticket.
• Мне нужна ручка. — I need a pen.
• Мне нужно место. — I need a place.
• Мне нужны две штуки. — I need two things/pieces.

So нужны is plural because there are two of them.

Because штука is a feminine noun.

Russian uses two forms of two:

• два with masculine and neuter nouns
• две with feminine nouns

Examples:

• два стола — two tables
• два окна — two windows
• две книги — two books
• две штуки — two pieces/items

Since the counted noun here is штука, and штука is feminine, you must use две.

After the numbers 2, 3, 4, Russian nouns usually take a special form that looks like the genitive singular.

So:

• одна штука
• две штуки
• три штуки
• четыре штуки
• but пять штук

This is part of a broader counting pattern in Russian:

• 1 → nominative singular
• 2–4 → genitive singular form
• 5+ → genitive plural

So две штуки is the normal correct form.

Штука is a very common, fairly informal word meaning something like:

• thing
• item
• piece
• one of those

In counting, it is often used when:

• the exact noun is obvious from context,
• the speaker does not know or does not want to repeat the noun,
• or they are talking casually.

So две штуки can mean:

• two items
• two pieces
• two of them

Its tone is usually conversational. It is not rude, but it is less precise than naming the object directly.

For example:

• Мне нужны две чашки. — I need two cups.
• Мне нужны две штуки. — I need two of them / two items.

Usually, no. That sounds incomplete or unnatural in most situations.

Russian usually wants either:

• the actual noun: Мне нужны два билета
• or a counting word like штуки if the noun is omitted: Мне нужны две штуки

Also, два by itself does not work well here because the speaker normally needs to indicate two of what.

If the context is extremely clear, native speakers may shorten things in conversation, but for a learner, две штуки is much safer than trying to say just два.

Yes, it is natural in everyday speech, especially when the object is already understood from context.

For example, in a shop, if you are pointing at something, you could say:

• Мне нужны две штуки.

That would sound like:

• I need two of these.
• I’ll take two.

However, if you know the noun, it is often more natural and clearer to say it:

• Мне нужны две батарейки.
• Мне нужны две тарелки.

So the sentence is natural, but a little vague on purpose.

Yes. Russian word order is flexible.

The neutral order here is:

• Мне нужны две штуки.

But depending on emphasis, you could also hear:

• Две штуки мне нужны.
• Нужны мне две штуки.

These all mean basically the same thing, but the emphasis changes.

For example:

• Мне нужны две штуки. — neutral
• Две штуки мне нужны. — emphasizes the number two
• Нужны мне две штуки. — more marked, often emotional or contrastive

As a learner, the original order is the safest and most natural default.

Then both the number and нужен would change according to the gender of the noun.

Since штука is feminine:

• Мне нужна одна штука. — I need one item/piece.

Compare with other genders:

• Мне нужен один билет. — masculine
• Мне нужна одна ручка. — feminine
• Мне нужно одно место. — neuter

So with one, Russian shows gender very clearly in both the number word and the needed word.

The patterns are:

• Мне нужны три штуки.
• Мне нужны четыре штуки.
• Мне нужно пять штук.

A very important point: with numbers like 5, 6, 7..., Russian often uses нужно rather than нужны in this construction.

So learners often notice this contrast:

• Мне нужны две штуки.
• Мне нужны три штуки.
• Мне нужны четыре штуки.
• Мне нужно пять штук.

That is because larger numerals behave differently grammatically in Russian.

For your sentence, though, with две штуки, нужны is correct.

Not always.

Sometimes штука really does mean thing/item/piece, especially when counting. But it can also have other uses depending on context, such as:

• referring vaguely to an object,
• meaning a gadget/device,
• even referring informally to a trick or interesting feature.

In your sentence, it is specifically a counting word for objects.

So here it is best understood as:

• two items
• two pieces
• two of them

rather than the broad English word thing in every possible sense.