O.K. My history is not as good as it should be, and this is a complicated topic. I am sure I am going to make some mistakes, so I am hoping djrak or Ghazar will see this and correct me wherever needed.
The Armenian Orthodox Church basically has four patriarchs at the following sees: Etchmiadzin, Cilicia, Jerusalem, and Constantinople. The patriarchs of Etchmiadzin and Cilicia each have the title of "Catholicos." The fact that there are four patriarchates is, in part, due to the Armenian nation's rather chaotic history.
The four patriarchs all belong to the same Armenian Orthodox Church. So it is not like there are four churches, rather it is just one Church with four patriarchs.
The Catholicos of Etchmiadzin is the first and "universal" patriarch. That means the others have jurisdiction over limited geographical areas, but the Catholicos of Etchmiadzin has jurisdiction everywhere else. That is the see which was established by St. Gregory the Illuminator in 301. Etchmiadzin is located in what is now the Republic of Armenia.
About a thousand years ago or so, the part of the Armenian homeland where Etchmiadzin was located was invaded. I think it was invaded by Mongols. The Catholicos of Etchmiadzin had to flee and he went to Cilicia. Eventually, Etchmiadzin was safe from the invaders and the Catholicosate there was restored. However, Cilicia had had a catholicos for so long, that a separate Catholicosate was kept there, and that is the origin of the Catholicosate of Cilicia. During the Genocide of 1915, the Armenians living in Cilicia were slaughtered or driven from their homes by the Turks. As a result, the Catholicos of Cilicia is now located in Antelias, Lebanon.
It is my understanding that the Armenian Patriarch of Constantinople was set up at the beginning of the Ottoman Empire and his jurisdiction is over Armenians living in Turkey.
I really don't know much about the history of the Armenian Patriarch of Jerusalem, except that the Armenian community over there is quite ancient. The Patriarch of Jerusalem, obviously has jurisdiction over the Armenians living in Israel.
Like I said above, it is a complex historical issue and I am sure I left out lots of details and got some things wrong. Djrak and Ghazar, if you read this, please fill in the blanks!