When I press 'download' for a song for example
The media player opens automatically and plays the song
And there is no download
Who can I do to make the download??

Really depends on the site hosting the media.
Some sites are setup to stream rather than download the tracks.

Just to provide more context for your question, you're using Windows, correct? Also, what are you downloading the music files from and if it's the internet, what browser are you doing so in?

You can try right clicking on it and choosing "Save Link As" in Firefox or "Save Target As" in IE.

Sometimes your browser automatically will play certain media right in it, with the right plugins.