Leaving out what's obvious
An ellipsis is a sentence with parts missing because the situation or the previous sentence supplies them. In conversation the full sentence often sounds laboured. The skill lies in keeping exactly what is new.
The same order
Complete
Io prendo un caffè, e lei prende un cappuccino.
I'll have a coffee, and she'll have a cappuccino.
Correct, but heavy.
Elliptical
Io, un caffè. Lei, un cappuccino.
A coffee for me, a cappuccino for her.
How it sounds at the bar.
Ordering and asking
When it is clear what should happen, naming what you want is enough. The person comes first, then after a short pause the thing.
Per me, una birra media.
A medium beer for me.
Il conto, per favore.
The bill, please.
Un attimo, arrivo.
Just a moment, I'm coming.
Answering briefly
English short answers lean on an auxiliary: Yes, I am. I did. Italian has no such tag and echoes the main verb or the participle instead. After a question word, the requested part alone is enough, together with its preposition.
Question and answer
A
Vieni?
Are you coming?
B
Sì, arrivo.
Yes, I'm on my way.
A
Hai capito?
Did you understand?
B
Capito.
Got it.
A
Con chi esci stasera?
Who are you going out with tonight?
B
Con Giulia.
With Giulia.
A
Ti è piaciuto il film?
Did you like the film?
B
Moltissimo.
Very much.
sì and no instead of the verb
When two things are contrasted, sì and no replace the whole verb. After verbs of thinking and hoping they take di.
Io sì, lui no.
I do, he doesn't.
Il vino sì, la birra no.
Wine yes, beer no.
Domani piove? – Spero di no.
Is it going to rain tomorrow? – I hope not.
Not repeating the verb
In a series with the same verb, you say it only once. This ellipsis is normal in written Italian too, often with a comma where the verb would be.
Io prendo la pasta, lei il pesce.
I'll have the pasta, she'll have the fish.
Marco lavora a Roma, sua moglie a Milano.
Marco works in Rome, his wife in Milan.
Ordering and answering ellipses belong in conversation and chats. In a formal email you write full sentences, although brief acknowledgements like Ricevuto, grazie are common there.