East Orthodox Christianity and Roman Catholicism are two out of three major christian denominations (third being protestantism).
These two churches are product of the Great East-West Schism of 1054. The schism was result of many political, cultural and some theological differences between Rome and Constantinople (between Pope of Rome and Patriarch of Constantinople). The culmination was when Pope Leo IX sent letter to Patriarch of Constantinople Michael Cerularius (who had closed all Latin churches in Constantinople year before) in which he cited Donation of Constantine (a fake document in which Emperor Constantine I transfers all political and religious power to Bishop of Rome i.e. Pope). Patriarch dismissed the document and Pope sent legates to Constantinople, but when legates came to Constantinople they were not received kindly and to make things worse Pope died soon after that resulting in even more complications which finally led to mutual excommunication and separation to Eastern Orthodox Church and Roman Catholic Church.
The main difference between Orthodox and Catholic Christians today is that Orthodox do not recognise Pope as the head of all Christians. Furthermore there is no unique central government in Orthodox Church i.e. most of the orthodox Churches adopt national title like Russian Orthodox Church, Ukrainian Orthodox Church, Serbian Orthodox Church etc. And each of these Churches has their own Primate (patriarch, archbishop...).
There are also some minor differences regarding liturgy and sacraments. For example Orthodox priests are allowed to get married (with exception of bishops) while all Catholic priests must practice celibacy. Regarding sacraments the greatest difference is about confession which is viewed much differently by Catholics and Orthodox (only Catholics have the type of confession we usually see in movies). Also it's worth mentioning that some of the Orthodox Churches use still Julian calendar. There are also some differences between fasting and generally about food - you can learn more by following the links below.
And at last one small difference which you would be hard to find in books or even on the Internet. When making a sign of cross Catholics move hand up-down-left-right while Orthodox move hand up-down-right-left.
There are many other smaller or grater differences between Orthodox Christianity and Catholicism and I could write about it whole day and still not say everything there is to say about the topic. Therefore you should check some of the links in the text and/or those below the text.