die Uni
university
Colloquial short form for Universität (university), an institution of higher education with the right to award doctorates.
Sie studiert Medizin an der Uni in Heidelberg.
She studies medicine at the university in Heidelberg.

university
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university
Colloquial short form for Universität (university), an institution of higher education with the right to award doctorates.
Sie studiert Medizin an der Uni in Heidelberg.
She studies medicine at the university in Heidelberg.

'Uni' (university) is the common and widely used short form of the word 'Universität' (university). In spoken language, 'Uni' is almost always used.
Meine Schwester fängt im Oktober an der Uni an.
My sister is starting university in October.
To describe the place where someone studies, you use 'an der Uni' (at the university). When entering the building, you say 'in die Uni gehen' (to go into/to the university).
Er hat seinen Master an der Uni Hamburg gemacht.
He did his master's at the University of Hamburg.
Common expressions related to 'Uni' (university) are 'an die Uni gehen' (to go to university/start studying), 'von der Uni fliegen' (to be expelled/removed from university), and 'die Uni schmeißen' (to drop out of university).
Nach dem Abitur wusste er nicht, ob er an die Uni gehen sollte.
After finishing secondary school, he didn't know whether he should go to university.
university
'die Universität' and 'die Uni' usually mean the same thing in everyday language. 'Uni' is the short, colloquial form of 'Universität'. In conversations with friends or in everyday situations, 'Uni' sounds very natural. In official texts, applications, or formal situations, 'Universität' is better. Both can mean the institution itself, and often also the place where you have lectures.

Mein Bruder fährt im Zug zur Uni und liest noch seine Notizen.
My brother takes the train to university and reads his notes on the way.
Lukas besucht morgen die Universität, weil er sich für ein Studium interessiert.
Lukas is visiting the university tomorrow because he is interested in studying there.