First off - being a widow in the early Church, as in Judaism, was considered to be both a disadvantageous situation (ie, they often had to live off alms or their kids’ support) and a position that was specially loved and protected by God. Even the pagan Romans thought of widows as a symbol of virtue; a chaste and reserved widow was an “univira” (a woman of only one husband) despite all the problems involved with being alone.
A widow who remained faithful to her dead husband (and thus protected her kids’ inheritance from Dad, and prevented those nasty stepfamily problems common in the ancient world) was in a position to be able to pray more, and to be heard by God. And indeed, a lot of saints are widowed women.
Therefore, there was such a thing as the “order of widows” in the early Church. Like vowed virgins, the widows funded by the Church (and sufficiently old enough to be inspirational and not tempting) were specially blessed/consecrated to their service. As a sign of this, they sat up front in the women’s section.
(And actually, some bishops still consecrate widows, albeit it’s pretty obscure in the West.)
Widowers didn’t get the props and goodies, but it was generally acknowledged that a “husband of one woman” was superior in spiritual status (assuming all else was equal) to a man who remarried. This shows up in Paul’s comments to Timothy on who would make a good bishop.
On the other hand, it was perfectly normal for a widow or widower to remarry, during most periods of the Church, especially if it was done for the sake of giving children a provider or caretaker.
There were a few times when remarriage after a spouse’s death was frowned on, however. If you read the Fathers, a lot of them felt that if God took your spouse, obviously He wanted you to be a widow or widower, and that it was grabby to get remarried as a matter of course, or just because you wanted sex. (And to be fair, a lot of young widows and widowers were unfairly pressured by their families to remarry, so defending widowed singleness was important.) So there was a good deal of argument over whether second marriages ought to be allowed to Christians; or if it were just a nice form of serial polygamy, like pagan/Jewish divorce and remarriage.
But Paul’s comments about “It’s better to marry than burn” have won out, overall.