Ellipsis
Japanese leaves out whatever is already clear from context: the subject most of all, but also objects and whole phrases. A conversation is often verbs and particles with everything else understood.
The subject disappears
English needs a subject in every sentence, even an empty one ("it rains"). Japanese does not. Once it is clear who you are talking about, you stop saying it. Starting every sentence with 私は (as for me) sounds like a textbook, not a person.
行く？ 行く！
Reading: いく？ いく！
Going? Going! (Are you coming? Yes, I am.)
食べた？
Reading: たべた？
Did you eat? (No subject, no object.)
明日、来る？
Reading: あした、くる？
Coming tomorrow?
Objects and places go too
Anything that was just mentioned, or that both of you can see, is dropped, and its particle goes with it.
読んだ？ うん、読んだ。
Reading: よんだ？ うん、よんだ。
Read it? Yeah, I did. (The book from the sentence before.)
駅はどこですか？ あそこです。
Reading: えきはどこですか？ あそこです。
Where is the station? Over there. (Not "the station is over there".)
When to put it back
Say the subject when it changes, when there is a contrast, or when you introduce someone new. That is what は and が are for; see the topic on は versus が.
私は行くけど、彼は行かない。
Reading: わたしはいくけど、かれはいかない。
I am going, but he is not. (A contrast: both subjects stay.)
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