Like every language, English has its own collection of words to express family relationships. There are the easy ones like mother, father, brother, and sister. Also comparatively easy are cousin, aunt, uncle, niece and nephew. But most people don't know about this "removal" part, beyond the fact that your "first cousin twice removed" is somebody you probably met once at a wedding ten years ago.
It's really not that complicated. Some people think removal has to do with estrangement, but actually it has to do with generations. The children of your Nth cousin are your Nth cousins once removed. Their grandchildren are your Nth cousins twice removed. In general, to find your Nth cousin K times removed, go to your Nth cousin and then follow their children K times.
You can also look at it from the point of view of the younger relative: Your Nth cousin K times removed is the Nth cousin of your Kth direct ancestor, where K = 1 is your parent, K = 2 is your grandparent, etc. For example, your father's second cousin is your second cousin once removed. There's a nice chart at genealogy.com that depicts many of these more complicated relationship terms.
In practice, these relationship terms are considered unnecessarily formal. In casual conversation, two such relatives who are comparable in ages are called distant cousins. If one is significantly older than the other, then the older relative is called a distant aunt/uncle of the younger one.
In many cultures, the terms for aunt and uncle can be applied more generally to any close friend of your parents. When I mentioned to my mother that the children of one of my friends call me Uncle Raymond, she asked me, "Which uncle?" This question puzzled me initially, but then I realized that in my parents' native language, as with most other Chinese dialects, there are several types of family relationships which in English all get lumped together as uncle. The term to use depends on whether the uncle is paternal or maternal, whether the uncle is older than or younger than your parent, and whether the uncle is by blood or by marriage. For example, you can use the word for husband of father's older sister or, if you wanted to be more respectful or if you need to resolve ambiguity, you can say the word that means husband of father's older sister number two. One of the reasons for so many fine distinctions for family relationships is that it is considered disrespectful to address someone older than you by name. (Compare United States culture, where it is considered disrespectful to address one's parents by name.) Consequently, resolving ambiguity cannot be accomplished as it is in Western cultures by appending the relative's name (Aunt Carol or Uncle Bob); it must be done by using a more specific title.
But since my friend's children speak English, they can just address me as Uncle and not have to worry about which kind of uncle I am. (To my Chinese-speaking nieces, however, I am 大姑丈.)