Frequency Response of Circuits - Laplace Transforms

I'm a second year Electrical Engineering student, and my lecturer recently introduced some concepts which I haven't been exposed to before relating to the Laplace transform (the course is on the frequency response of circuits). I've only got a superficial understanding of Laplace transforms.

My main question is there appears to be some ability to use the Laplace transform of a function to break it into transient and steady-state parts. How is this?

Also, when I learned Laplace transforms, we didn't discuss what 's' actually is. What is it? Apparently it's some complex number (sigma + omega * i) and, when we're concerned with the steady-state response, it equals (omega*i). What is going on here?

I'm guessing I'm not coming across clearly - that's because I don't really know where to start. I'm a bit lost. If someone could clarify the topic in general (frequency response of circuits, with particular reference to Laplace transforms and representation in the Laplace domain), that would be awesome.

I'm more than happy with being pointed to good reference material rather than being told the answers. If it's a simple explanation though, it'd be nice to be handed it straight-up in an answer.

I don't know about circuits, but I believe you're trying to figure out the "physical meaning" of the Laplace transform. If you have any experience with the Fourier transform, you'd know that it analyzes the signal in terms of sinosoids, while the Laplace transform analyzes the signal in terms of sinousoids and exponentials. Traveling along a vertical line in the s-plane reveal frequency content of the signal weighted by exponential function with exponent defined by the constant real axe value. Traveling along a horisontal line reveal the exponential content of the signal weighted by sinousoids function with frequency defined by the constant imaginary axe value. In particular, traveling along the imaginaly axe reveal frequency content of the signal weighted by 1 which is equivalent to the Fourier transform of the signal. In general the laplace variable $s=\sigma +j\omega$ with $σ=0$ is equal to the Fourier transform of the signal. I believe that this textbook might be helpful: http://www.amazon.com/Circuits-Signals-Systems-William-Siebert/dp/0262192292