Router musí být připojený tak, aby fungoval internet.

Literally, it means The router must be connected.

The structure is:

• Router = router
• musí = must / has to
• být = to be
• připojený = connected

So Czech is using must + be + connected, just like English.

Because the sentence is describing a state, not the action itself.

• být připojený = to be connected
• připojit = to connect

So Router musí být připojený means the router needs to be in the connected state.

If you used připojit directly, the meaning would shift toward the action of connecting, not the resulting condition.

Because it agrees with router.

Router is:

• masculine
• singular
• nominative
• inanimate

So the adjective/participle also takes the masculine singular nominative form:

• připojený

If the noun were feminine, you would get something like připojená. If it were neuter, připojené.

tak, aby means so that, in such a way that, or so that the result is...

In this sentence:

• tak = so / in such a way
• aby = so that

Together, they introduce the desired result:

• Router musí být připojený tak, aby fungoval internet.
• The router must be connected so that the internet works.

The word tak adds the sense of properly / in the right way / in such a way.

Yes.

You could also say:

• Router musí být připojený, aby fungoval internet.

That still means The router must be connected so that the internet works.

Adding tak makes it feel a bit more like:

• connected in the right way
• connected in such a way that...

So tak adds nuance, but it is not absolutely necessary.

Because aby introduces a subordinate clause.

In Czech, subordinate clauses are normally separated by a comma, so:

• Router musí být připojený tak, aby fungoval internet.

That comma is standard and expected.

This is one of the most common questions learners have.

After aby, Czech usually uses a form that looks like the past tense form, but here it is not actually past tense.

So:

• funguje = works / is working
• aby fungoval = so that it works / so that it would work

In other words, after aby, Czech uses a special clause type that often corresponds to English so that, in order that, or sometimes a kind of subjunctive-like meaning.

So fungoval here does not mean worked in the past. It means something like work in a purpose/result clause.

The subject is internet.

So:

• fungoval internet = the internet worked

Even though the verb comes before the noun, internet is still the subject.

Czech word order is much more flexible than English, so the subject does not always have to come before the verb.

Often, yes.

Both are correct:

• aby internet fungoval
• aby fungoval internet

The version with internet before the verb is often a bit more neutral.
The version in your sentence puts internet at the end, which can give it a little more emphasis.

So the given sentence is grammatical, but the word order is partly about focus and style.

It agrees with internet.

Here internet is masculine singular, so the form is:

• fungoval

Compare:

• masculine singular: fungoval
• feminine singular: fungovala
• neuter singular: fungovalo

So the ending reflects the gender and number of the subject.

Because Czech has no articles.

There is no direct equivalent of English the or a/an in normal Czech grammar.

So:

• Router can mean the router or a router
• internet can mean the internet

The exact meaning depends on context.

Yes, very often.

You may also hear or read:

• Router musí být připojen tak, aby fungoval internet.

Both připojený and připojen are possible here.

A rough difference is:

• připojený = more everyday, adjective-like
• připojen = often more formal, technical, or concise

In many real-life situations, both are natural.

Not necessarily.

musí means must / has to, but in a sentence like this it usually expresses practical necessity, not a command from a person.

So here it means something like:

• The router has to be connected this way if the internet is going to work.

It is closer to a technical requirement than an order.

In modern Czech, internet is commonly written with a lowercase letter when it means the general network.

So:

• internet = standard modern spelling in ordinary usage

You may still see Internet sometimes, especially in older texts or certain styles, but lowercase internet is very common today.