Correct spelling, explanation: the word siege originates from Old French siege, sege, and from Vulgar Latin sedicum. As we can see, all the previous forms were spelled with -i as the first vowel and -e as the second vowel, not the other way around. The word seige doesn’t exist in any dictionary. The only correct version is siege.

Definition of siege:
noun, surrounding with the army some areas, fortresses, and cities in order to capture them
The siege of the fortress was successful.
I’m sure you heard about that siege. It lasted almost four months.

Collocations with siege:
Some most commonly used collocations include: the siege of the city/castle/base, put an end to the siege, call off the siege, and to be under siege.

Incorrect spelling, explanation: the word seige is just an error because nothing in its etymology indicates that it should be spelled like that. If we look at its origins, we can clearly see the specific order of vowels: -i first, then -e, not the other way around. Therefore, seige is an incorrect form and the correct one is siege.