The new series of Doctor Who started in 2005 and brought back a lot of the mythology, monsters, and ideals of the original series. The new run follows the adventures of the ninth, 10th, and 11th incarnations of the Doctor.
Although it is a continuation of the classic series, the viewer does not need to see the earlier episodes to understand and enjoy this new installment of the science fiction show. The experience of watching the new seasons would be enhanced by knowing the basics of the original run, but it can definitely stand alone.
You can explore the three new variations of the Doctor in more detail under the tabs listed as "Ninth Doctor," "Tenth Doctor," and "Eleventh Doctor."