If my x wife moves in a boyfriend to her home, do I still have to pay her alimony?

Depends on your State and what your Divorce Decree says.

You can always go back to Court and attempt to get the Decree/Order changed.

Is it written as alimony or spousal support?

For future reference please choose the forum more carefully. Legal questions belong in the law forums where this has been moved to.

Generally, yes. Most divorce decrees include under what conditions alimony stops. Usually there is either a set period or until remarriage. So read the decree. It might allow you to ask for a modification if circumstances change.

I know in Ohio the woman can live with a boyfriend for as long as she wishes and still collect her court ordered alimony. The alimony stops only when the court ordered timeframe runs out or she gets remarried. There are numerous women who skirt around this alimony problem by just living with a boyfriend whether it is in her home or his home. Happens all the time and is perfectly legal. Only if pertinent language was incorporated into the divorce decree about alimony payments specifying that the woman can't reside with a male as either a boyfriend or a roommate the alimony must be paid.