"Do you love me?"
Translation:Ты меня любишь?
Меня is the accusative form of я, so it's used when the first person is the direct object of a sentence. (Do you love me?) Мне is the dative form, and is used when the first person is the indirect object of a sentence. (Send me the letter.) (In constructions like мне нравится, the ся at the end of the verb is the reflexive particle—cf. себя 'one/him/her-self', which takes the place of the direct object, and makes the speaker the indirect object.)
Of course, the accusative and the dative have other functions, but that's the basics.
It has to do with emphasis. The nearer a word is to the start of a sentence, the more important it is. Ты любишь меня: "You like me, but do you love me?" Ты меня любишь: "Do you love me?"
I think I have that the right way around, anyway. My professors in Russia focused much more in the intonation patterns than on the word orderings.
The way I understand it is that if there is no specific intonation for emphasis in Russian, then the word that comes last in the sentence is the "news."
Ты её любишь = You love her. "Love" is the "news" in the statement.
Ты любишь её = You love her. Whom do you love? Her.
Of course, vocal emphasis supersedes the word order.