أضع الحقيبة قرب الحاسوب في المكتب.

أضع is the imperfect (non-past) verb form, 1st person singular: I put / I place.

• Root idea: وضع (to put/place)
• أضعُ = I put / I am putting / I usually put (context decides)

In Arabic, the subject pronoun is often built into the verb. The prefix أ- in أضع signals I. So you normally don’t need أنا unless you want emphasis:

• أنا أضع الحقيبة... = I (as opposed to someone else) put the bag...

ال- makes the noun definite: the bag (a specific, known bag).
If you mean a bag (indefinite), you would say something like:

• أضع حقيبةً قرب حاسوبٍ في مكتبٍ (fully vowelled/formal)
  In real unvowelled writing, people often just write: أضع حقيبة قرب حاسوب في مكتب and rely on context.

It is the direct object of أضع (the thing being placed). In fully vowelled MSA it would be:

• أضعُ الحقيبةَ
  because direct objects are typically accusative.

قرب means near / close to. It’s a common, neutral way to express proximity.
You could also use alternatives with slightly different feel:

• بجانب = beside / next to
• بالقرب من = near to (very common; explicitly uses من)

Both patterns exist in MSA:

• قربَ الحاسوبِ (often treated like a preposition-like word; the next noun is genitive)
• بالقربِ من الحاسوبِ (explicitly uses من)
  Your sentence uses the first, shorter pattern: قرب الحاسوب.

With الحاسوب, you’re saying the computer (a specific one, presumably in that office).
If you meant any computer (indefinite), you’d use حاسوب (and in fully vowelled MSA it would usually be حاسوبٍ after قرب).

في المكتب means in the office and adds the broader location.
Arabic word order is flexible; you can often move prepositional phrases for emphasis or flow, for example:

• أضع الحقيبة في المكتب قرب الحاسوب.
  Both are acceptable; the meaning stays basically the same.

Yes, in fully vowelled formal MSA you’d typically see:

• أضعُ (final ـُ for indicative mood)
• الحقيبةَ (accusative direct object)
• قربَ الحاسوبِ (often: قربَ
  • genitive after it)
• في المكتبِ (في takes genitive)
  In everyday writing and speech, these endings are usually not written or fully pronounced.

Common ones:

• ق in قرب: a deep q sound (back of the throat), not English k.
• ح in الحقيبة / الحاسوب: a “breathy” h from the throat (stronger than English h).
• ض in أضع: an emphatic d-like sound; it “darkens” nearby vowels.