To show possession in Hungarian, you just add on suffixes in the same way that you would with the typical cases. There are some exceptions, which I will explain to the best of my ability.

I cover plural possession in this post.

If you haven’t already, you should go over Vowel Harmony and Vowel Lengthening, so you can know how to apply the correct endings.

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Possession: Singular Objects

Én

• words ending in a vowel: -m
• back-vowel words: -om (sometimes -am¹)
• front-vowel words: -em
• rounded front-vowel words: -öm

a macska – a macskám
the cat – my cat

az anya – az anyám
the mother – my mother

a szék – a székem
the chair – my chair

a hörcsög – a hörcsögöm
the hamster – my hamster 

¹ “a” as a linking vowel: this was briefly touched upon in my post about plurals, and it applies here with possessives as well. Most back-vowel words will take “o” as a linking vowel, but some words take “a” instead. Unfortunately there doesn’t seem to be a simple, clear-cut rule on this, and the only thing I can suggest is memorizing the words that take “a” as you come across them.

For example, “my house” (ház) would be “házam” and not “házom”.

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Te

• words ending in a vowel: -d
• back-vowel words: -od (sometimes -ad¹)
• front-vowel words: -ed
• rounded front-vowel words: -öd

a kutya – a kutyád
the dog – your dog 

az ágy – az ágyad
the bed – your bed

a feleség – a feleséged
the wife – your wife

a hörcsög – a hörcsögöd
the hamster – your hamster

¹ “a” as a linking vowel: same situation as above.

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Ő  (Ön)¹ 

(¹Ő (xe) and Ön (you – formal sg.) use the same endings)

• words ending in a vowel (back-vowel word): -ja
• words ending in a vowel (front-vowel word): -je
• back-vowel words: -a
• front-vowel words (including rounded): -e

NOTE: There are actually special rules for the 3rd singular/2nd person singular formal possessive ending. Alex was kind enough to write an explanation here (downloadable PDF file). While it specifically says 3rd person, it applies to Ön/Önök as well, as they take the same endings as ő and ők respectively.

¹The Hungarian 3rd person singular “ő” is not gender-coded and can mean he/she/they(singular). To be inclusive and also practical, I use “xe” as opposed to “he/she/they(sg.)”, because it’s rather cumbersome and time-consuming to type out.

Xe = he/she/they(sg.), xir = his/hers/theirs(sg.), xem = him/her/them(sg.).

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Mi

• words ending in a vowel: -nk
• back-vowel words: -unk
• front-vowel words (including rounded): -ünk

a kutya – a kutyánk
the dog – our dog

az autó – az autónk
the car – our car 

az ablak – az ablakunk
the window – our window

a gyerek – a gyerekünk
the kid – our kid 

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Ti

• words ending in a vowel (back-vowel words): -tok
• words ending in a vowel (front-vowel words): -tek
• words ending in a vowel (rounded front-vowel words): -tök
• back-vowel words: -otok (sometimes -atok¹)
• front-vowel words: -etek
• rounded front-vowel words: -ötök

¹ “a” as a linking vowel: same situation as the 1st and 2nd person above.

a macska – a macskátok
the cat – your (pl.) cat

a körte – a körtétek
the pear – your (pl.) pear 

az erdő – az erdőtök
the forest – your (pl.) forest

a polc – a polcotok
the shelf – your (pl.) shelf 

a szék – a széketek
the chair – your (pl.) chair

a hörcsög – a hörcsögötök
the hamster – your (pl.) hamster

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Ők (Önök)¹

(¹Ők (they) and Önök (you – formal pl.) use the same endings)

• words ending in a vowel (back-vowel words): -juk
• words ending in a vowel (front-vowel words): -jük
• back-vowel words: -uk
• front-vowel words (including rounded): -ük

NOTE: There are special rules for the 3rd person plural/2nd person plural formal as well. You can read Alex’s explanation here (downloadable PDF file).