If you fight occasionally, that's nothing to worry about; everybody gets into it once in awhile. If there are a few things about you that bother him (or her), though, and he or she's ALWAYS bringing them up, or he/she simply can't stop putting you down, now there is a problem. If he/she continually insults you, if he/she rarely compliments you, if he/she doesn't treat you the way you want to be treated-- dump him/her like the garbage he/she is. It doesn't matter how much he/she "cares" if he/she's always putting you down, because if he/she's putting you don't he/she doesn't really care. Don't be fooled by false tears.
Don't let him or her bullshit that he/she's a better person than you. It doesn't matter who he/she is, or who you are-- if you're in a loving relationship together, you're equals. No one should keep score on who's beating who. If he/she parades himself as superior, if he/she acts like he/she's too good for you-- let him/her go; he/she's not worth it. He/she doesn't love you; the person loves himself/herself, and he/she raises himself higher by pushing you down. Give this loser a reality check: do yourself a favor and break up with him/her.
Deal with name-calling the same way. If he/she acts like a jerk, he/she obviously doesn't care. No truly loving boyfriend or girlfriend seriously calls his or her boyfriend or girlfriend "bitch" or "slut" or "ho." Let him/her go, and make that decision final. When you trash him/her he/she may try to be sweet, begging for you to come back, promising he/she'll change, but don't fall for the lies. The only people who can really treat you right would never do that kind of stuff in the first place.
The same technique applies for jealousy: you should be able to date your boyfriend or girlfriend and still talk to other people. You should be able to get a hug from a male or female friend without him or her getting angry. You should be able to go somewhere without having to tell him or her exactly where you're going. He/she shouldn't accuse you of cheating without good cause. He/she shouldn't try to control your every move. If he/she can't handle being without you, he/she can't really handle being with you, either. This person doesn't trust you if he/she always has to grill you with twenty questions. But relationships are built on trust, so if this person doesn't trust you, you really don't have much of a relationship, at least from his/her side. He/she doesn't "love" you; he/she loves being in control. Drop your chains and get rid of him/her.