Taylor series and radius of convergence

I have a couple of interrelated questions about series:

Although I've used series for some time, I never really understood what the range of convergence really means. If I have a Taylor series of the function Exp(x) around x=0, does that mean that if I take into consideration only the first 3 terms of the series, it will be very close to the original function around x=0, and that the more terms I take into consideration of the series, the polynomial generated by the series will be getting closer and closer to the real function, both around x=0 and also farer and farer away from that point?

This question is related too to asking how does a graphic calculator calculate what sin(0.5) or cos(0.75) values are. Could anyone explain? I'd like something more visual or not too deep on maths, if possible(Sleepy).

Also, what does it mean to say that the series is centered around x=0, or x=5?