Alignment of text in itemize environment

Another option would be to use the enumitem package to define a new list having the desired layout (the showframe option for geometry was only used as a visual guideline):

\documentclass{article}
\usepackage{enumitem}
\usepackage[showframe]{geometry}

\newlist{aims}{enumerate}{1}
\setlist[aims,1]{
  label={Aim~\arabic*},
  leftmargin=*,
  align=left,
  labelsep=10mm,
  itemindent=\dimexpr\labelsep+\labelwidth+7pt\relax
}

\begin{document}

\begin{aims}
\item This aim fits nicely on to a single line.
\item The details of this aim are such that they will not fit on to a single line. Now the text aligns with the margin.
\item This aim fits nicely on to a single line.
\item The details of this aim are such that they will not fit on to a single line. Now the text aligns with the margin.
\end{aims}

\end{document}

Since I wasn't sure about the desired alignment, here's another possibility:

\documentclass{article}
\usepackage{enumitem}
\usepackage[showframe]{geometry}

\newlist{aims}{enumerate}{1}
\setlist[aims,1]{
  label={Aim~\arabic*},
  leftmargin=*,
  align=left,
  labelsep=10mm,
}

\begin{document}

\begin{aims}
\item This aim fits nicely on to a single line.
\item The details of this aim are such that they will not fit on to a single line. Now the text aligns with the margin.
\item This aim fits nicely on to a single line.
\item The details of this aim are such that they will not fit on to a single line. Now the text aligns with the margin.
\end{aims}

\end{document}