Ég geymi bankakortið í vasanum.

Geyma means to store/keep/save (something in a place)—often implying you put something away somewhere.
geymi is the present tense, 1st person singular form:

• infinitive: að geyma
• present: ég geymi, þú geymir, hann/hún/það geymir, við geymum, þið geymið, þeir/þær/þau geyma

A rough pronunciation guide (varies by accent):

• Ég: like yeh(g) (the g is often very soft)
• geymi: roughly GAY-mi (with an Icelandic ey diphthong)
• bankakortið: BAN-ka-kor-tið (final ð is like the th in this, often very soft)
• í: like ee (long i)
• vasanum: VA-sa-num (stress on the first syllable, as usual in Icelandic)

It’s the direct object of geyma, so it’s in the accusative. For many neuter nouns like bankakort, nominative and accusative look the same, so you often tell from the sentence role rather than the form alone.
Paradigm (singular, definite) for a typical neuter like bankakort:

• nominative: bankakortið
• accusative: bankakortið
• dative: bankakortinu
• genitive: bankakortsins

Both can be correct but they mean slightly different things:

• í vasanum = in the pocket (a specific pocket, or the pocket in the current context—often your pocket)
• í vasa = in a pocket (more indefinite/generic)

Because after í (in/into), Icelandic uses:

• dative for location (in/at, not moving)
• accusative for motion (into, movement toward)

Here it’s location (the card is already there), so vasanum is dative definite singular of vasi (pocket). The -num is a common dative definite ending for masculine nouns.

Then you would use accusative after í:

• Ég set bankakortið í vasann. = I put the bank card into the pocket.
  Here vasann is accusative definite singular (motion into).

The dictionary form is vasi (masculine). A common singular pattern is:

• nominative: vasi / definite vasinn
• accusative: vasa / definite vasann
• dative: vasa / definite vasanum
• genitive: vasa / definite vasans
  (There are variations across nouns, but this is a typical masculine pattern.)

The given order is very natural: Subject – Verb – Object – Place.
You can move the place phrase for emphasis:

• Ég geymi bankakortið í vasanum. (neutral)
• Í vasanum geymi ég bankakortið. (emphasizes in the pocket)
  But note that when a phrase comes first, Icelandic often keeps verb-second structure in main clauses.

Common options:

• Ég geymi bankakortið í vasanum mínum. = in my pocket (explicit)
• Ég geymi bankakortið í vasanum. often already implies “my pocket” from context, so the possessive is frequently omitted unless you need contrast or clarity.

Geyma is good if you mean store/keep (in a specific place). Depending on nuance, you might also hear:

• Ég er með bankakortið í vasanum. = I have the bank card in my pocket (very common, focuses on “having it on me”)
• Ég held bankakortinu í vasanum is less idiomatic for this meaning; halda tends to mean hold (in your hand) or keep/maintain in other senses.