The Aliens over-all goal is to take over the Earth, they don't actually want to destroy humanity, as they wish us for experimentation, food and slaves. Actually, the Aliens are genetically related to Earthly lifeforms! Untold millenia ago, they visited our world, perhaps they brought life itself, certainly they altered or interbred with Terran life.
  Humans' aggressive tendencies make us ideal candidates for becoming the next generation of Alien combat troops, like the Floaters or Mutons!
  The Ethereals are the leaders of the Aliens, but even they have a leader... On Cydonia, a region on Mars, they have a huge base, here is the master of the Aliens: a massive cybernetic computer.
  Their goal is to get rid of organized human resistance, to do this they plan to  terrorize the populace, and/or infiltrate the upper echelons of society.  By kidnapping humans, and either psionically, cybernetically or even genetically modifying them, they use political and social leaders (such as rock stars), to carry the "New Faith", that of Alien and human peace! While the Aliens have problems understanding our society, as we do theirs, they can see ways humans can be influenced. They're not above bribery either, if that's what it takes.
  When the Aliens "take over" a country, all funding for XCOM from that region stops, however XCOM can probably still operate there, helped by those opposed to the invaders.
  The Aliens' main problem is that Mars and Earth, even for them are far apart, and they need resources, making it difficult to carry out their plans, ergo they try to build Alien Bases on Earth.
  XCOM is their main enemy, hence they will try to locate XCOM bases to overwhelm them, and stir up anti-XCOM feelings when they can. 

  Every three months, the Aliens will try to construct a base on Earth, shortening their supply lines and aiding their cause.
  A typical Alien Base will have 9 Small, Medium and Large Scout craft, as well as 3 Abductors and Harvesters, 2 Terror Ships and a Battleship. Hidden by a variety of means, they are nearly impossible to locate, unless an XCOM craft can track a Supply Ship UFO to the exact region and patrol for a few hours. the Aliens may decide to blow the intruder away before they can learn too much...Alien bases swarm with horrors. Typically they have around 3 times as many aliens as a battleship carries, if not more! Hence it's not wise for XCOM to assault an Alien Base until  very well armed and equipped.
  Bases may be in the Antartic, buried into a volcano's flanks or in the depths of a jungle. They will be spread out over several levels, but they do have much in the way of items that XCOM would like, such as Elerium 115, Alien Alloys etc.