Genealogy is research into your own ancestry. Specifically, it can be a case of simply discovering names, relationships and dates. More generally, it can be the study of the social and economic history of your family. There are lots of approaches:
The classic approach is to follow one line of your ancestors. For reasons connected with the way early records were kept, this is often the male line i.e. your father, his father, and so on.
Discovery of all your ancestors. Harder – because (a) as noted above lines through females tend to be more difficult to find; and (b) each time you go back one generation, you’ll have (approximately) twice the number of people to locate.
Finding all descendants of a given ancestor. Also hard. Everyone had parents even if we can’t discover their identity; not everyone had children. Everyone had two parents (neither more nor fewer); people have varying numbers of children.
A one name study. Some genealogists are so fascinated by their surname that they record all occurrences of it with the aim of assigning them to family groupings (at least) and, ideally, of discovering how they are all related. Unless the surname is unusual and typically associated with one part of the country only, the latter may not be possible. Finding that someone is carrying out a one name study on a family you are interested in is, however, a great advantage – a lot of the work will have been done for you already!
Family history. One or some combination of the above plus personal details about your ancestors. In practice, it is almost impossible to compile a pedigree of your family without discovering other details – in particular, addresses and occupations. However, you’re likely to discover lots of other details too – in particular in wills. However, records such as those kept by the armed forces about those who served with them may shed interesting light on your ancestors too.