The unsrtnat style orders the bibliography by citation order. Similar to the example from LaTeX/BibTex not arranging citations by order of appearance:
\documentclass{article}
\usepackage[semicolon]{natbib}
\begin{filecontents}{\jobname.bib}
@book{foo,
author = {Foo},
year = {2002},
title = {Title of Foo},
}
@book{bar,
author = {Bar},
year = {2005},
title = {Title of Bar},
}
@book{baz,
author = {Baz},
year = {2008},
title = {Title of Baz},
}
\end{filecontents}
\begin{document}
\section{The Beginning}
First we cite \citep{foo}.
Now we cite \citep{bar}.
Then we cite \citep{baz}.
Finally, we cite \citep{foo,bar,baz}.
\bibliographystyle{unsrtnat}
\bibliography{\jobname}
\end{document}
I take it that by "Harvard style" you mean a generic author-year style ffor citation call-outs. The harvard package was among the first to implement such a style for LaTeX users. By now, there are quite a few other packages as well that provide possibilities for generating author-year citation call-outs.
To instruct the natbib citation management package to use an author-year rather than a numeric citation style, it's best to load the package with the authoryear option:
\usepackage[authoryear]{natbib}
Assuming you have an entry in your bib file with a key of "bloggs:2012" that is authored by "Jane Bloggs", you could issue the command
\citep{bloggs:2012}
to generate a "parenthetical" citation reference of (Bloggs, 2012). To get a "textual" citation, viz., Bloggs (2012), using the same entry, you'd type
\citet{bloggs:2012}
The precise manner in which the bibliographic entries will be typeset in the References section depends critially on the bibliography style file you use. You may want to start with the plainnat.bst and see if this meets your needs. I.e., issue a command such as
\bibliographystyle{plainnat}
somewhere in your document.
Finally, to have LaTeX make each citation into a hyperlink to the corresponding item in the bibliography, you need to load the hyperref package, preferably with the options colorlinks=true and citecolor=<MyColorOfChoice>.
Addendum, prompted by the OP's provision of an MWE. Alright, I understand now that you aren't using BibTeX for now to construct the bibliography but, instead, are building the bibliography "by hand" by assembling a bunch of \bibitem entries.
There are two issues with your MWE. First, if you're going to do the \bibitems by hand, the \bibitem in question must look like this:
\bibitem[Bloggs(2012)]{bloggs:2012} Bloggs, J., The Journal of Stuff, 2012.
Without any material in square brackets, LaTeX is going to use the default citation call-out style, i.e., generate numeric labels when it encounters a citation of this bibitem.
Second, don't type
Something is true (Bloggs, 2012)\citep{bloggs:2012}
in the text. Rather, type something like
Something is true \citep{bloggs:2012}.
and let LaTeX provide the text of the citation call-out -- (Bloggs, 2012) in this case. If the hyperref package is loaded, the string (Bloggs, 2012) will automatically be made into a hyperlink to the corresponding entry in the references section.
That said, I would strongly recommend that your learn at least the basics of how to use BibTeX. Doing so will save you a lot of time in the future.
The fully revised MWE would look like this:
\documentclass[11pt]{article}
\usepackage[authoryear,round]{natbib}
\bibliographystyle{plainnat}
\usepackage[colorlinks=true,citecolor=blue]{hyperref}
\begin{document}
\section{First Section}
Something is true \citep{bloggs:2012}. As argued by \citet{bloggs:2012}, \dots
\begin{thebibliography}{99}
\bibitem[Bloggs(2012)]{bloggs:2012} Bloggs, J., The Journal of Stuff, 2012.
\end{thebibliography}
\end{document}
Best Answer
Natbib does not "do all the work": it depends on loading an appropriate bibliography style. It is the style (not natbib) which is responsible (with bibtex) for the detailed format of entries in the bibliography, and for sorting the bibliography. Natbib is not "tied" to any particular bibliography style, though it does "come with" some basic styles with which it is compatible. But there are other styles available, depending on your detailed needs.
For your purposes it looks to me (as it does to egreg and others) that the style you want is
plainnat. Far from being incompatible withnatbibit was intended for it, as the name suggests. which will produce an alphabetically sorted bibliography. The reason you are seeing the (undesired) sorting behaviour is that you are using a style which is intended to produce references in order of citation.So try changing your file to
\bibliographystyle{plainnat}. Delete your .aux, .blg, and .bbl files, and re-run LaTeX -> bibtex -> LaTeX -> LaTeX. This should produce a bibliography ordered as you want it to be. If you are getting errors in this, it's certainly not for any incompatibility withnatbib; but tell us what problems you are seeing, and we can try to help.