Learn Russian: Это книга сестры.

Here, Это means "This is" and acts as a generic subject. When you are introducing or pointing something out with "This is...", you always use the neutral Это, regardless of the gender of the noun that follows. You would only use the feminine Эта if you were saying "This book" (Эта книга).

Why does the English translation say "my sister's" when there is no word for "my" in the Russian sentence?

In Russian, it is very common to leave out possessive words like "my" when the meaning is obvious from context. You are allowed to say Это книга моей сестры, but usually, if you just mention a family member, listeners will naturally assume you mean your own.

Why is the word for sister сестры instead of сестра?

сестры is in the Genitive case, which is used here to show possession (just like adding an apostrophe "s" in English). For feminine nouns ending in -а, like сестра, the ending changes to -ы to show ownership. So книга сестры literally translates as "book of the sister".

Is the word order fixed? Could I say Это сестры книга instead?

In Russian, the standard word order for possession is to put the owned object first, followed by the owner. So книга сестры is the most natural way to say it. Saying сестры книга is grammatically possible but sounds unusual and poetic, so it is best avoided at this stage.