I suggest you to read some articles about this subject. Smashing magazine has a great post about it, to see which one is right for you. They have some definitions about these layouts and I believe they are very accurate:
Fixed:
A fixed website layout has a wrapper
that is a fixed width, and the
components inside it have either
percentage widths or fixed widths. The
important thing is that the container
(wrapper) element is set to not move.
No matter what screen resolution the
visitor has, he or she will see the
same width as other visitors.
Fluid:
In a fluid website layout, also
referred to as a liquid layout, the
majority of the components inside have
percentage widths, and thus adjust to
the user’s screen resolution.
Elastic:
There is a third option when working
with Web page layouts. An elastic
design is sometimes preferred by
designers because it mixes the two
other main layout types. It works by
sizing all elements with em’s. The
quote below explains exactly what an
em is and why it can be beneficial.
The pros and cons in the article are very concise too, with examples and further researching.
If you choose to use "fluidic" layouts, read their article about the subject as well.