You've got to remember in the UK, DW is iconic. Quite a few of us grew up with it over the course of the 1963-1989 era of the show. So when it returned we didn't need a massive introduction to it, for kids watching it, it might be a case of ask your parents.
The overall plot is fairly simply.
It's about an alien who travels time and space usually with a companion or two (not always human), fighting evil and injustice.