Yes, Christopher Eccleston was the 9th Doctor. Before 2005, Doctor Who didn't have "series" which is the equivalent of "seasons" here in America, they were just episodes. So, "Series 1" refers to the modern Doctor Who shows that started with Eccleston in 2005. We are now in the 7th Series and the first half of it ends next Saturday 9/29. After that there will be a "Christmas Special and then the series will resume sometime after.
Maybe after you watch the modern DW, you can start learning and watching some of the "classic" (1963-1989) DW with the 1st to 8th Doctor.