During my class in Istanbul I´ve learned "Bu bir çiçek." I wanted to review some basic Turkish just now, using an online application, and it said "Bu bir çiçekter". What is the difference, and which one is better... and why?

Im the native Turkish and according to me, there is no difference between them and if i were you, i would say "Bu bir çiçek", if you say " Bu bir çiçektir" nobody would misunderstand you in here.

-dir/-tir suffix is used in some informative contexts.

...-dir/-tir = Be informed that ... or I inform you that ..., Just in case you may not know about it, I let you know that ..., etc.

Okay, so the most ´normal´ way would be to just say, for example, "siyah cüzdan." And if somebody asked "what colour is the wallet?" you could say "cüzdan siyahtır." ...or basically just say "siyah cüzdan" everytime?

I think you could say "cüzdan siyah" in case if you mean "wallet is black".

The meaning of these two words can be completely different depending on their order:

siyah cüzdan - Black wallet (siyah is an adjective here)

cüzdan siyah OR cüzdan siyahtır - Wallet is black (siyah and siyahtır mean "is black" here). As far as I know, the difference between "siyah" and "siyahtır" here is that "siyahtır" is more formal, and in everyday´s speech people rather use form without "tir/dir/tır/dır" suffix (so they probably would rather say "cüzdan siyah" than "cüzdan siyahtır" - but see si++´s explanations above).

Generally, if you place adjective to the left of a noun, then it works only as a description (attribute) of a noun:

Büyuk ev - Big house

O büyük evde yaşıyorum - I live in that big house.

Büyük bir evde yaşıyorum - I live in a big house (<- I´m not sure if it could be said like "Bir büyük evde yaşıyorum" ???)

If adjective is placed to the right of a noun, then it works as a verb "is ..." :

Ev büyük - House is big

O ev büyük - That house is big

Yaşadığım ev büyük - House in which I live is big (is a big one?)

In that case, you can also add -tir/-tır/-dir/-dır suffix to make it more formal:

Ev büyüktür - House is big

O ev büyüktür - That house is big

Yaşadığım ev büyüktür - House in which I live is big (is a big one?)

Using -dir/-tir usually implies some information being passed to the other parties. If absent it usually reflects the opinion of the speaker.

Atatürk büyük bir lider = (In my opinion) Atatürk is a great leader

Atatürk büyük bir liderdir = (I declare that or it should be known that) Atatürk is a great leader