Churches and other religious congregations tend to sort themselves along demographic lines, so it's very common to have congregations that are mostly one specific group of people (not just by race, also often by, say, country parishioners' ancestors are from, or specific religious movement within a religion, etc.) Even within congregations, there are often multiple services that people might choose to go to based on these things. Many of these subgroups have long cultural traditions, and may have different musical traditions, favorite holidays, and important traditions that aren't shared outside of that group.

However, that's almost always based on preference, and on where people feel comfortable, culturally accepted, and understood. While some places that really do only want people from specific groups to attend do exist, they're usually easy to identify, and they won't invite other people to their events (most other congregations will). Even those places will usually be polite to an obvious visitor, although they might suggest a different congregation to attend in the future.

This is not to discount the long history of segregation within some denominations that led to the creation of, for instance, the African Methodist Episcopal Church, or of "black" and "white" congregations in the same denomination in the same community, but today if you show up to most US churches they would welcome -superficially at least- anyone of any race. You'll often get a more effusive welcome if you're from the demographic that doesn't usually attend that service, since people know that you're visiting and want to make you feel welcome. Over a longer period time, the social expectations may not be very accepting of people not in the majority group, which enforces the tendency for people to gravitate towards the groups where they're in the majority- and it's very common in the US for people to "shop" for a religious congregation where they feel comfortable, often across denominational or even major theological lines.