I am a beginner. I learnt that "Je suis un homme" translates to "I am a man", but to say "I write", I should use "J'écris". Is "j'" a contracted form of "je" like "don't" for "do not"?
J' is a Je indeed, but it is not the same as the do not/don't contraction.
In the case of do not/don't, you contract because you want to save some effort/time by biting one syllable.
In the case of je/j', you don't get to choose whether you contract or not. It is a matter of pronunciation: when je is followed with a word beginning with a vowel, instead of pronouncing the e then the vowel, you eliminate the hiatus by removing the e.
In formal language, this is the only case when the e should disappear. In spoken French (or transcriptions, for books for instance), it is sometimes ignored nonetheless, and phrases like je suis are often pronounced chuis and transcripted as j'suis.
Yes, "J'écris" is the contraction of je and écris. Unlike in English, contracted forms in the French language are mandatory.