What Manufactorer makes the best Cat Back exhaust system for a 2003 350Z. Please give me some suggestions.

What do you mean by best?
There are a lot of "tier 1" or "high end" exhaust systems.

Do you mean best as in "best deal" or best as in performance? Best bang for buck?

nismo stillen GReddy those are just some

Bang for the Buck:

Rs*R
Invidia
Injen

I have the Rs*R and it's a nice setup. If you look hard, you will find one for about $800. It eliminates the 60lbs of stock exhaust and replaces it with a nice Y pipe and two straight mufflers. Nice aggressive rumble, but I typically keep the silencers in . . . Only problem I know of is that the Y pipe is mild steel and will corrode over time. I may have it refinished to prolong its use.

Apexi N1, All Greddy and Invidia N2 sound really good and get rid of the low exhaust tones at CRUISING speeds. Do some research on MY350Z.com and you will find dynos, prices and sound clips.