eksi is an abbreviation for "Enna koduma sir idhu".
The orginal dialogue was first used in the rajinikant starrer Chandramukhi and has been parodied ever since, most notably by premji in the tamil movie chennai-28.
Since then the phrase has become a frequenter in many tamil conversations and was subsequently abbreviated and moved in to the sms lingo. It is usually used to express displeasure/unhappiness about something, showing some signs of grief and sometimes surprise.
"Oh god! look at the blood oozing out! Eksi"
"he likes that ugly girl! Eksi"