I noticed that when I got Owain to join my entourage, he had the same hair color as my character. Are there other effects that come about by different marriages?
The characters in Fire Emblem: Awakening are divided into 1st generation and 2nd generation, aka. "parents" and "children".
Each child is bound to a female parent character (the mother). There are only two children that do not adhere to that rule: Lucina, daughter of Chrom, and Morgan, daughter of the male Avatar. You can only get the children if the mother (or father, in the case of Lucina and female Morgan) is married.
Children inherit a couple of attributes from their parents, according to Serenes Forest:
Base stats and growth rates are inherited, but these aren't very important in the long run. Weapons and items are not inherited, since the parents are still around.
Children characters inherit the sum of their parent's maximum stat modifiers, +1. If your Avatar pairs with a children character, Morgan does not receive the +1 to his/her modifiers.
The children inherit the last active Skill from each of their parents. This can include Skills from gender-specific classes (except Special Dance), so you can have Demoiselle on male children. Chrom's children are exceptions--his daughter(s) will always learn Aether and his son will always inherit Rightful King (even if Chrom hasn't learnt either Skill). Skill inheritance occurs at the moment you enter the mission required to recruit the child, so you've got plenty of time to play around with the parents' Skills.
In general, children inherit all the regular class options from both their parents (so a lot of classes if one of their parents is your Avatar), except classes exclusive to the opposite gender. If a class option cannot be inherited due to gender issues, that option will be replaced with another option more appropriate for their gender. Despite being a Special class, Villager can be inherited by male children and both Taguel and Manakete can be inherited by Morgan.
Classes and Skills obtained from DLC (this includes Skills learned by DLC classes) can not be inherited.
Additionally, a child will always inherit the father's hair color.
Anna, Say'ri, Tiki, Flavia, Emmeryn and Aversa do not have children, but can be paired up with the Avatar to produce Morgan. Children characters do not produce any offspring, except if paired up with the Avatar to produce Morgan.
I'm not positive what you mean by "other effects" but as far as children's look is concerned.
The child is based of the mother. For example, this means that Sumia will ALWAYS have Cynthia, regardless of the father. The possible exceptions are Chrom (who will always have Lucina and the mothers child) and Avatar, who if male can have Morgan and the mothers child.
The child's hair will be the color of the fathers. The one exception here is female Morgan (male Avatar's child) who will have her mothers hair.
Parents also affect the classes and stats of children, but that is a little more in depth. If you would like to read about it please visit this site and scroll down to the Awakening section.