Heute öffne ich nur einen Browser, weil mein Computer langsam ist.

In a normal German main clause, the finite verb must be in second position. This is often called the V2 rule.

So when Heute is placed first for emphasis, the verb öffne must come next:

• Heute öffne ich nur einen Browser.

The subject ich comes after the verb here.

Compare:

• Ich öffne heute nur einen Browser.
• Heute öffne ich nur einen Browser.

Both are correct, but the second one gives today more emphasis.

Because Browser is the direct object of öffne, it must be in the accusative case.

Browser is a masculine noun, so the indefinite article changes like this:

• nominative: ein Browser
• accusative: einen Browser

Since the sentence means I open a browser, the browser is the thing being opened, so it takes the accusative:

• Ich öffne einen Browser.

You learn the grammatical gender with the noun:

• der Browser = the browser

That tells you it is masculine. Once you know that, the article changes depending on the case:

• der Browser (nominative)
• den Browser (accusative with the)
• ein Browser (nominative with a)
• einen Browser (accusative with a)

Nur means only or just.

In this sentence, it most naturally modifies einen Browser:

• Heute öffne ich nur einen Browser
  = Today I’m opening only one browser

It suggests limitation: not two browsers, not many tabs/programs, just one browser.

Word position can slightly affect emphasis, but here the meaning is clear.

Because weil introduces a subordinate clause, and in German subordinate clauses are separated from the main clause by a comma.

So:

• Heute öffne ich nur einen Browser, weil mein Computer langsam ist.

That comma is required in standard German.

Because weil creates a subordinate clause, and in German subordinate clauses the conjugated verb usually goes to the end.

Main clause:

• Mein Computer ist langsam.

Subordinate clause with weil:

• ..., weil mein Computer langsam ist.

So the basic pattern is:

• weil + subject + other elements + verb

This is one of the most important word-order patterns in German.

Mein Computer ist langsam is the normal word order for a main clause.

But after weil, it becomes a subordinate clause, and that changes the word order:

• main clause: Mein Computer ist langsam.
• subordinate clause: ..., weil mein Computer langsam ist.

So the difference is caused by weil, not by the words themselves.

Mein Computer is in the nominative case because it is the subject of the subordinate clause:

• weil mein Computer langsam ist

The computer is the thing that is slow, so it is the subject.

Also note that mein works like an article here, so you do not use another article:

• mein Computer
• not der mein Computer

Because langsam is being used as a predicate adjective after the verb sein (to be).

In German, adjectives used after verbs like sein, werden, and bleiben do not take adjective endings.

So:

• Der Computer ist langsam.
• Der Browser ist neu.
• Das Auto ist schnell.

But before a noun, the adjective does take an ending:

• ein langsamer Computer

No. Heute can appear in different places depending on emphasis and style.

Examples:

• Heute öffne ich nur einen Browser.
• Ich öffne heute nur einen Browser.

Both are correct. Putting Heute first emphasizes today more strongly.

German is flexible, but the verb-second rule in main clauses still applies.

It is capitalized because it is the first word of the sentence, not because it is a noun.

Normally, heute is written with a lowercase h inside a sentence:

• Ich öffne heute nur einen Browser.

In German, all nouns are capitalized, but heute is not a noun here.

The base form is öffnen, meaning to open.

Here it is conjugated for ich in the present tense:

• ich öffne
• du öffnest
• er/sie/es öffnet
• wir öffnen
• ihr öffnet
• sie/Sie öffnen

So öffne is simply the first-person singular present form.

Yes. In German, einen Browser öffnen is natural and means to open a browser.

You may also hear similar expressions like:

• ein Programm öffnen = to open a program
• eine Datei öffnen = to open a file
• einen Tab öffnen = to open a tab

So this is a normal and idiomatic use of öffnen.

Yes. Browser is a loanword used in German, especially in technology contexts.

In German it is treated like a normal noun:

• der Browser
• ein Browser
• einen Browser

And like all German nouns, it is capitalized.

Yes, but the meaning changes.

• einen Browser = a browser / one browser, not previously specified
• den Browser = the browser, a specific browser already known from context

So:

• Heute öffne ich nur einen Browser = I’m opening only one browser today
• Heute öffne ich nur den Browser = I’m opening only the browser / that specific browser today

The sentence you were given uses einen because it is talking about a browser in a general sense.