Type in console as root:
apt-get update && apt-get install php5-curl
or with sudo:
sudo apt-get update && sudo apt-get install php5-curl
Sorry I missread.
1st, check your DNS config and if you can ping any host at all,
ping google.com
ping zm.archive.ubuntu.com
If it does not work, check /etc/resolv.conf or /etc/network/resolv.conf, if not, change your apt-source to a different one.
/etc/apt/sources.list
Mirrors: http://www.debian.org/mirror/list
You should not use Ubuntu sources on Debian and vice versa.