How to Install & Enable Curl on Ubuntu LAMP Stack?

You only have to install the php5-curl library.
Alternative 1
You can do this by running

sudo apt-get install php5-curl

You will need to restart the server afterwards:

sudo service apache2 restart

Alternative 2

sudo apt-get update
sudo apt-get install php5

sudo apt-get install libapache2-mod-php5

sudo apt-get install curl libcurl3 libcurl3-dev php5-curl

and then:

cd /etc/php5/apache2/

sudo nano php.ini

and add this command: extension=curl.so
the last thing restart apache.

Source: http://askubuntu.com/questions/9293/how-do-i-install-curl-in-php5