Daj mi odvijač kojim mogu da zategnem vijak.

Because mi is the unstressed dative form of ja and means to me.

In this sentence, the speaker is the person who should receive the screwdriver, so Serbian uses the indirect object:

• Daj mi odvijač = Give me the screwdriver / Give the screwdriver to me

Mene is a different form of ja and would not be used for this role here.

It is changing case, but for a masculine inanimate noun like odvijač, the accusative singular looks the same as the nominative singular.

So:

• odvijač = nominative singular
• odvijač = accusative singular

That is normal in Serbian for masculine inanimate nouns.
The same thing happens with vijak in this sentence.

Compare that with a masculine animate noun:

• čovek → vidim čoveka

It is a relative clause describing which screwdriver the speaker wants.

So the sentence does not mean just any screwdriver. It means:

• the screwdriver with which
• or that I can use to tighten the screw.

In other words, the clause narrows down odvijač.

Kojim is a form of the relative pronoun koji.

Here it agrees with odvijač:

• odvijač is masculine singular
• so the pronoun is also masculine singular

But its case is determined by its role inside the relative clause, not by its role in the main clause. Inside the clause, it means with which, so it appears in the instrumental:

• kojim = instrumental singular of koji

So:

• odvijač = the noun being described
• kojim = with which

Because Serbian often uses the instrumental case by itself to express the tool or means used for an action.

For example:

• Pišem olovkom. = I write with a pencil.
• Sečem nožem. = I cut with a knife.

The same idea appears here:

• kojim mogu da zategnem vijak = with which I can tighten the screw

Using sa kojim is usually less natural here. For tools and instruments, the bare instrumental is the normal choice.

Because in modern Serbian, after many verbs, especially modal verbs like moći, the very common pattern is:

• da + present tense

So:

• mogu da zategnem = I can tighten

This is one of the most important Serbian sentence patterns.
English often prefers an infinitive, but Serbian very often prefers da plus a present-tense verb form.

This is a question of aspect.

• zategnem = perfective
• zatežem = imperfective

Here, the speaker wants a screwdriver for a single complete result: tightening the screw. That is why zategnem is natural.

Very roughly:

• zategnem = tighten it successfully / all the way / as one complete action
• zatežem = be tightening / tighten repeatedly / focus on the process

In this sentence, the perfective verb fits better because the goal is completion.

Yes, it is morphologically a present-tense form, but after da and a modal like mogu, it expresses a possible or intended action.

So:

• mogu da zategnem does not mean I am tightening right now
• it means I am able to tighten / I can tighten

Also, perfective present forms in Serbian often do not describe an action happening right now. They commonly refer to a complete action viewed as possible, future, or expected in context.

The sentence has a very natural, neutral word order, but Serbian word order is somewhat flexible.

A few things matter:

1. mi is a clitic, so it normally comes very early in the sentence.
2. The relative clause usually comes right after the noun it describes:
   • odvijač kojim mogu da zategnem vijak

So this version is the most natural:

• Daj mi odvijač kojim mogu da zategnem vijak.

You can move things around for emphasis, but not every rearrangement sounds good. Learners should treat this sentence as a very solid default model.

No. There are common alternatives, depending on region and level of formality.

For odvijač:

• odvijač = standard
• šrafciger = very common colloquial word

For vijak:

• vijak = standard/technical in many contexts
• šraf = very common in everyday speech

So in casual speech, you may also hear:

• Daj mi šrafciger kojim mogu da zategnem šraf.

That does not change the grammar pattern; it is mostly a vocabulary difference.