How to get part name in LaTeX ?

Putting the part title into the header is not common, therefore there are no default macros for this.

You could redefine the \part macro to save its argument to e.g. a \parttitle macro which then can be used in the custom defined headers:

\newcommand*\parttitle{}
\let\origpart\part
\renewcommand*{\part}[2][]{%
   \ifx\\#1\\% optional argument not present?
      \origpart{#2}%
      \renewcommand*\parttitle{#2}%
   \else
      \origpart[#1]{#2}%
      \renewcommand*\parttitle{#1}%
   \fi
}

That would depend on what type of "part" you are refering to, \chaptername, \sectionname, \thepage, \thesection, \thechapter, etc... See this link for details.

EDIT: In light of your comment, this thread seems to be discussing a solution (note: there is a typo, it must be \renewcommand{\part}..., i.e. a backslash must be added):

For using the parts, you can do the following:

\let\Oldpart\part
\newcommand{\parttitle}{}
\renewcommand{part}[1]{\Oldpart{#1}\renewcommand{\parttitle}{#1}}

and then use \parttitle in the header.

Another alternative is to use the titlesec package which provides commands like \partmark that you can use.

In the KOMA-classes \part executes a \partmark command which you could use to set either \rightmark or \leftmark to the part title.

But as \part always generates a new page and you probably need the header only on the next page you can also do something simple like this:

\newcommand\partcontent{}
...

\part{Blub}
\renewcommand\partcontent{Blub}

and then use \partcontent in the header.