"Ailem benim için önemli."
Translation:My family is important for me.
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This explains it well. If something has emotional value to you (your family, best friend, pet, family home) it is important to you. So usually a person or, if an object, a treasured one.
Important for you also means something you value, but without emotions being involved eg "It is important for me to pay my bills on time" or "It is important for me to have a mobile phone for my work." So usually a desirable situation or, if an object, a functional one.