It would be difficult to overestimate the influence our ancestors have on us today! It is to them that we owe our health, hair and eye color, body shape and proportions, talents, and personalities. Yet few of us really know our ancestors, and that is especially true if we are adopted.
If we are fortunate, we will know our parents. We will also know our siblings, if any. They form the nuclear family that we were raised with, and who had the most nurturing influence on us. We may not get along with them all the time, but they are family and they will always be there. The women in our lives may have married, but we always refer to them by their maiden names (the name they were given at birth) when doing family history or genealogy.
If we are adopted, our birth records may be sealed but that is changing in many states and countries. It may take time, money and connections, but many adoptees are now able to access the records of their births. Birth records, however, may have limited information such as only the mother's name. An adopted child may never know his or her father's name but, with the mother's name and the date and place of the child's birth, an adopted child is usually able to trace his maternal line.
The next generation back consists of our grandparents. We may have known them - perhaps even been raised in some part by them. They are the parents of our parents. Many times a grandchild will see traits and characteristics of the grandparents repeated by their parents and, to their chagrin, may see their ancestors in themselves! The children of our grandparents, and the siblings of our parents, are our aunts and uncles. They will also likely have married and had children - our cousins. If you are LDS and doing temple work for the dead, please remember that we are to perform these ordinances only for our direct-line ancestors, and then only with permission from the living next-of-kin.
If we count ourselves as the first generation, the fourth generation back includes our great grandparents. Some have been fortunate enough to have known and remember some of this generation. I was 18 years old when my last living great grandfather died. Unless they lived with or near your family while you were young, you may not know them well but my great grandfather lived with us for several months each year while I was young. A lumberjack by trade, then a farmer, he had great stories to tell. He taught me to whistle and I introduced him to the books about Paul Bunyan. He also misled me (unintentionally, I hope) as to the origins of his ancestors! It wasn't until I was researching our ancestors in Canadian censuses that I found out that Granddad's grandfather was one of 13 John Bakers living in the same village. Twelve of them came from Pennsylvania but our ancestor emigrated from England! In fact, "English John Baker" is written on his tombstone!
The children of our great grandparents (and siblings of our grandparents) are our great aunts and uncles. Their children are our second cousins. As we go back each generation of ancestors, we add a number to designate their relationship to us. Thus, the parents of our great grandparents are our second great grandparents and their parents are our third great grandparents and so forth.
Many web sites will lead us to records through which we can research our ancestors but some, like OneGreatFamily and parts of FamilySearch and The Generations Network, will take us right into the lineage that has already been researched, and lay out our genealogy for us as far back as someone else has submitted it.