Quick answer
Ц ц = ts, one single sound. It is always hard, so ци is pronounced “tsy”.
Цц
Цц
ts
центрcentre
How to say it
English has the sound, just never at the beginning: say “cats”, keep the ts, and drop the rest.
центрcentre
цена́price
цветcolour
у́лицаstreet
ца́рьtsar
оте́цfather
Always hard
Even when an и follows, Ц stays hard: цирк sounds like “tsyrk”. That is why the spelling rules treat it separately from ч and щ.
The tsar
Царь comes from “Caesar”, the same root as the German “Kaiser”. The word travelled from Rome through Byzantium and kept its ts.
Typical mistakes
Reading центр as “sentr”
“tsentr”
Ц is never a plain s.
Splitting it into t + s
one sound
It is a single consonant, like the ts in “cats”.
Check yourself
Which word contains Ц?
центр — centre.