This is because of the way \cvcomputer is defined. From cvmodern.cls:
% usage (inside 'computer skills' cvsection environment): \cvcomputer{category}{programs}{category}{programs}
\newcommand*{\cvcomputer}[4]{%
\cvdoubleitem{#1}{\small#2}{#3}{\small#4}}
So let's see how \cvdoubleitem is defined:
% usage: \cvdoubleitem{subtitle}{text}{subtitle}{text}
\newcommand*{\cvdoubleitem}[4]{%
\cvline{#1}{\begin{minipage}[t]{\doubleitemmaincolumnwidth}#2\end{minipage}%
\hfill%
\begin{minipage}[t]{\hintscolumnwidth}\raggedleft\hintfont{#3}\end{minipage}\hspace*{\separatorcolumnwidth}\begin{minipage}[t]{\doubleitemmaincolumnwidth}#4\end{minipage}}}
While I didn't bother to understand all of this, the minipages and a look at the CTAN template tells me that this type of entry is geared toward a two-column layout, presumably because the author assumed computer entries won't be very long.
If you don't want the two-column layout, I recommend redefining \cvcomputer like this:
\renewcommand{\cvcomputer}[2]{\cvline{#1}{\small#2}}
Make sure you remove the third and fourth arguments, which are empty, from your source. Your section would be like this:
\renewcommand{\cvcomputer}[2]{\cvline{#1}{\small#2}}
\section{Računalne vještine}
\cvcomputer{Operativni sustavi}{Windows, UNIX(Linux), izvrsno poznavanje rada operativnih sustava}
\cvcomputer{Programiranje}{Osnove rada u Phythonu}
\cvcomputer{WEB dizajn}{HTML}
\cvcomputer{Office paketi}{Znanje rada s Wordom, Excellom, Powerpointom}
\cvcomputer{Adobe}{Poznavanje rada s Adobe Photoshopom, Illustratorom i InDesignom}
\cvcomputer{Hardware}{PC, mreže}
BUT, why not make use of the two-column layout? Then, don't use the redefinition and just reorder the arguments:
\section{Računalne vještine}
\cvcomputer{Operativni sustavi}{Windows, UNIX(Linux), izvrsno poznavanje rada operativnih sustava}{Office paketi}{Znanje rada s Wordom, Excellom, Powerpointom}
\cvcomputer{Programiranje}{Osnove rada u Phythonu}{Adobe}{Poznavanje rada s Adobe Photoshopom, Illustratorom i InDesignom}
\cvcomputer{WEB dizajn}{HTML}{Hardware}{PC, mreže}
\homepage only takes one argument. So, you should just write:
\homepage{www.gmail.com}
You may want to use the template from CTAN, which can be found here, as it is the most up to date template.
It seems that the template that you're opening from the LaTeX Templates website automatically opens all of the associated .sty files in either of the online editors (i.e., sharelatex and writelatex), and it seems that that version of moderncv is version 1.1. The most recent version on CTAN, however, is version 1.5.1.
It seems that at some point (in particular, in version 1.1), \homepage actually could take two arguments, which is why this works on writelatex for you (i.e., because opening the moderncv template via the LaTeX Templates website automatically opens and uses the version 1.1 .sty files). However, this use seems to have been deprecated in (at least) version 1.5.1, which is the version you will have locally on your machine if your TeX distribution is up to date, since it is the most recent version on CTAN.
This change is not documented in the moderncv changelog, however, so it's not clear in which version that use of \homepage was deprecated.
Best Answer
This seems to a bug which was introduced by recent fixes:
For now you can revert this behavior by adding the following code block in your preamble:
EDIT: The bug will is fixed in an upcoming update of
moderncv. Meanwhile use the answer provided by @Xavier which is more advanced than my brute force fix.