First off, Dr Who regenerates, brilliant, which is why different actors across 60 years.
Last Dr was female (first female Dr) and she regenerated into this Dr which happens to be a past Dr that Donna was a traveling companion of. Rose, Donna's daughter name, happens to be the name of a traveling companion of this Dr's before Donna as well.
These are specials which happen around the holidays and I believe there is just one more with this Dr before he regenerates into the new Dr (Ncuti Gatwa) which will be the first Dr of color and possibly gay. He previously played a gay character in the show Sex Education (I found this show quite funny and it's on Netflix).
I believe they are doing these specials with this Dr (David Tenant) because his character with companions Donna and Rose really brought in a mass global viewership their first time around.
As for the episode itself, when we finished watching we both felt like someone had fast forwarded through some scenes (probably cut for time) so that probably didn't help new viewers like yourself.
