During the 12 months after burial, observant Jewish mourners avoid certain pleasure-seeking activities, like concerts with live music. For eleven months after the burial, the children of someone who has died say kaddish, a prayer sanctifying God's name, at services where there is a minyan, a quorum of 10 adult Jews. (In some synagogues, the adults must be male.) Some Jews who don't ordinarily pray in a congregation will go to services in order to say kaddish.
If you are a Jewish mourner who lives in a large or medium-sized Jewish community, you should be able to find a congregation that has daily morning services. In a place where there are fewer Jews, it may be more difficult to say kaddish everyday. Some people choose to go to synagogue on Shabbat once a week when that isn't their usual practice, or just say the prayer whenever they happen to have an opportunity. Even if you have not gone to synagogue regularly during the year or mourning, there is nothing wrong with saying kaddish on the one or two occasions a year you may happen to be at a synagogue.
For some people, the process of saying kaddish for a year is an opportunity to find comfort from other Jews who have been through the mourning process, since in many congregations, the people who make up the daily services do so because they benefitted from other people praying with them during the mourning year.
Though you should consult your own rabbi on this as on other issues of practice, most rabbis will say that Jewish children should say kaddish for non-Jewish parents. The only disagreement seems to be whether the Jewish child is permitted or obligated to do this. It may seem strange to use Jewish mourning practice to honor a person who was not Jewish, but the principle of honoring the dead person is the same.
The Guide to Death and Mourning for Interfaith Families is also available as a downloadable PDF and Word document.