I've made the switch from natbib to biblatex two years ago, so I should be able to answer this. That said, Seamus, Simon Byrne, and domwass have already made lots of good points.
(For anyone still asking "Why should I use biblatex?": See this answer [shameless plug].)
LaTeX document
With natbib, a model LaTeX document would look as follows:
\documentclass{<someclass>}
\usepackage[<options>]{natbib}
\begin{document}
A bare citation command: \citep{<key>}.
A citation command for use in the flow of text: As \citet{<key>} said \dots
\bibliographystyle{<somestyle>}
\bibliography{<mybibfile>}% Selects .bib file AND prints bibliography
\end{document}
With biblatex and its built-in styles, this changes to:
\documentclass{<someclass>}
\usepackage[<language options>]{babel}% Recommended
\usepackage{csquotes}% Recommended
\usepackage[style=<somebiblatexstyle>,<other options>]{biblatex}
% \bibliography{<mybibfile>}% ONLY selects .bib file; syntax for version <= 1.1b
\addbibresource[<options for bib resources>]{<mybibfile>.bib}% Syntax for version >= 1.2
\begin{document}
A bare citation command: \autocite{<key>}.
A citation command for use in the flow of text: As \textcite{<key>} said \dots
\printbibliography[<options for printing>]
\end{document}
Note that I used \autocite instead of \parencite which is the actual counterpart of natbib's \citep. \autocite is a high-level citation command that will be translated into the low-level bare citation command appropriate for the chosen style - e.g. it will enclose a citation in parentheses in authoryear styles, but produce a footnote citation in authortitle styles. Even more, it will automatically move trailing punctuation.
For some of the custom (not already built-in) biblatex styles, additional preamble adjustments may be advisable - see the example provided by Seamus for biblatex-apa.
As Simon Byrne has mentioned: If you don't want to change every instance of \citep and \citet in every document to its biblatex counterpart, use the natbib=true compatibility option.
Typically, you'll select one or several local .bib files as your bibliographic database; however, \addbibresource also allows to load remote resources and other data types (e.g., ris).
.bib file
domwass has already mentioned that changes to your .bib files are not mandatory, but you'll miss some of the goodies offered by biblatex. When I switched to biblatex, I changed my address fields to location and my journal fields to journaltitle. I also added hyphenation fields in order to be able to switch languages on a per-entry basis in the bibliography.
Biber
biblatex will work for the most part with traditional BibTeX and its 8-bit version bibtex8, but I recommend the use of Biber (the default backend since biblatex v2.0) for the following reasons:
Full unicode support.
No capacity issues. (In contrast, when using BibTeX with bibliographies of about one hundred entries, I've run into errors disguised as obscure warnings - see section 2.4.2 of the biblatex manual for details.)
Multiple or subdivided bibliographies will always be processed in a single pass.
Many biblatex features introduced since v1.1 (e.g., advanced name disambiguation, smart crossref data inheritance, configurable sorting schemes, dynamic datasource modification) are "Biber only".
Biber is included in TeXLive and MiKTeX; latexmk also supports the use of Biber.
texmf/doc/latex/biblatex/examples. – matth May 09 '13 at 08:28