The division comes, in a nutshell, from the human struggle of wanting to do what we wish or what we believe is right vs what the church tells us as Catholics. Some things are more black and white than others:
Divorce - "what God has joined let no man put asunder". Basically, God blesses a marriage through the priest or chaplain or deacon, etc and no man can change that.
Gays/lesbians - God created us in His image, plus humans are to marry and have sex and have kids; part of the commitment you make when you take vows. Gays and lesbians cannot do so. Also, this kind of sex is explicitly condemned in the bible in several places. It always comes up that "well, God made me this way, so it is OK". Well, no, it's not. There is a whole thread all of its own.
Sex - this is the area which causes the most stress, confusion, frustration, etc among Catholics. OK, causes a lot of it; depends on who and where you are on such matters. Some will say it is as simple as "the church says so, so that's it"; many will argue it is not that simple - and it is not. Bottom line is the church tells us what is and is not proper and allowed; some folks do not like that. Not just because it is human nature to want and have sex, but also because it is difficult to think of the church in the bedroom with you. This is all a very simplified answer, and only one incomplete one, to a very complicated issue.
Birth control - abortion is killing a human life, period; humans can only create humans, so a child is killed. As for birth control, God created us to love and procreate, and birth control prevents that. It puts pleasure first, or so goes the thought, without consideration for the "procreative and unitive" aspects of sex between couples. Yes, there is NFP which allows for sex and avoid pregnancy, but the couple is still "open to life", open to the possibility of pregnancy.
These are only quick answers and thoughts. Nothing is simple or only has one side. keep reading; I'm sure you will eventually be inundated.