Теперь система работает по-другому.

Here теперь means now in the sense of now compared with before or from this point on. It suggests a change from an earlier situation.

So Теперь система работает по-другому implies:

• before, the system worked one way
• now, it works in a different way

Сейчас also means now, but it often focuses more on right at this moment.
So:

• Теперь система работает по-другому = The system works differently now / now the system works differently
• Сейчас система работает... = more like At the moment, the system is working...

In this sentence, теперь is the more natural choice because the sentence is about a change.

Система is in the nominative singular because it is the subject of the sentence — the thing that is doing the action.

The basic structure is:

• система = the system
• работает = works / is functioning

So система работает means the system works / is functioning.

Russian does not have articles like a, an, or the.

So система can mean:

• a system
• the system

The exact meaning depends on context. In your sentence, English naturally uses the system, but Russian does not need a separate word for that.

Работает is the 3rd person singular present tense form of работать.

That means it matches система (system), which is singular:

• я работаю = I work
• ты работаешь = you work
• он / она работает = he / she works
• система работает = the system works

Because работать is an imperfective verb, the present tense can describe:

• what is happening now
• how something functions generally
• a current state or usual behavior

In this sentence, it means the system works / is functioning.

По-другому means differently, in another way, or not the same way as before.

It describes how the system works. So it is an adverb.

You can think of the sentence like this:

• система работает = the system works
• по-другому = differently

So the full sentence means The system works differently now.

It is written with a hyphen because Russian commonly forms adverbs with:

• по-
• plus a form ending in -ому or -ему

Examples:

• по-новому = in a new way
• по-старому = in the old way
• по-другому = differently / in another way

So the hyphen is standard spelling for this type of adverb.

Historically, that is how this kind of form developed, and you can still see that другому looks like the dative form of другой.

But in this sentence, it is best to learn по-другому as a single adverb meaning differently.

So in practical terms:

• it functions as one unit
• it answers the question how?
• it does not behave like a normal prepositional phrase you would translate word-for-word

In other words, do not read it too literally as according to another. Just understand it as differently / in another way.

Yes. Russian word order is fairly flexible, because the grammar is shown mainly through endings rather than fixed position.

The neutral order here is:

Теперь система работает по-другому.

But other orders are possible, for example:

• Система теперь работает по-другому.
• По-другому теперь работает система.

These versions are not identical in emphasis.

• Теперь система работает по-другому is a natural neutral statement.
• Система теперь работает по-другому puts a little more focus on the system.
• По-другому теперь работает система is more marked and emphasizes differently.

So yes, the order can change, but the original sentence is the most straightforward version.

No. Работать can mean several things, including:

• to work at a job
• to function
• to operate

In this sentence, it clearly means to function / to operate, because the subject is система (system), not a person.

So:

• Система работает = The system is working / functioning
• not The system has a job

Russian often uses теперь to imply change already, but if you want to make the contrast even clearer, context or extra words can do that.

For example:

• Раньше система работала иначе, а теперь работает по-другому.
  The system used to work differently, but now it works in another way.

But even without that extra context, Теперь система работает по-другому already strongly suggests that something has changed.