Divorces are discouraged, but allowed. 
 
Also this question is somewhat addressed here: http://en.fairmormon.org/Mormon_ordinances/Marriage/Jesus_said_that_divorce_not_allowed_except_for_fornication 
 
The divorce rate for Mormons is generally much lower than the general populace. For Mormons that marry in one of our temples the divorce rate is at least 50% lower than the national average. 
 
One of the major tasks of our Bishops (similar to a Pastor) is to counsel the members in his ward/congregation. He will spend a lot of time helping couples out. If it is too difficult a situation for him then we have professional marriage counselors available to help distressed couples.

Divorce is allowed but discouraged in the Church of Jesus Christ of Latter-day Saints.  
 
A unique aspect in the LDS Church is that when we marry in the Temple it is for time and eternity. So even when a couple is civilly divorced they are still married for eternity unless they request the removal of there temple sealing from the President of the Church.  
 
Usually the President of the Church will grant an unsealing when a civil divorce has happened.

I have a few family members of the mormon religion. I am wondering if there is any significance of a man re-marrying (for the 3rd time) on the day that his first wife (not of the mormon faith) died. please help me understand why this would occur and if it has anything to do with the mormon faith

Esther, 
 
Never heard of anything to relate this to the Church. There is nothing related to the church about this. Sounds like just some person's personal preference.