This entry should be a reminder to any LDS Church leadership reading this, and hopefully will provide some information for newly widowed members of the Church. If there is an error with any of this please let me know, but a most of the information I have on this I received after talking to a Temple President.
Widowed men in the Church can marry again, and have the choice of marrying for time and eternity in the Temple or they can marry for time only in the Temple.
To marry again for time and eternity both the man and woman must be worthy to enter and participate in Temple ordinances, and the woman must have never been married or has had any sealings from a previous marriage canceled by the appropriate authority.
To marry again for time only both the man and woman must be worthy to enter and participate in Temple ordinances, and the woman must be widowed and sealed to her deceased spouse.
Widowed sisters in the Church who are sealed to their deceased husbands can only marry for time only. This is scripturally based upon the Kingdom of God being a through a Patriarchal Line. Widowed sisters can marry for time only in the Temple, however the groom must be widowed as well. Divorced or never married men cannot marry for time only in the Temple.
Widowed sisters can marry for time only outside of the Temple with no limits to the groom's marital status, or membership in the Church.
The only exception to all of this is if the widow cancels the sealed marriage to her deceased husband. There are reasons why some sisters have chosen to do this, however it is extremely individualistic and will only be done with the Lord's blessing via the Church Leadership in Salt Lake.
This is probably worded better in the Church's Handbook of Instructions, in the section for the Bishop. If you have questions about this, speak with your Bishop or Branch President and have them look it up, or you can talk to the President of your local Temple if he is willing to discuss this with you.