It is not required to play an older game of the series to get to a new one as others said too,but note that it is all recorded in the history of the games.
What you do with your character in one game,might be written inside a book in the next game.
In the end all of the games say the same story: They record the history of the Empire and Nirn across the centuries.
Things happening.. Wars between the races,wars between factions,wars between gods and demons...
In one game you are some person but in the other you are another.
But that doesn't mean that your character from the past game never existed in the same world as your character in the new one...
Now allow me to tell you some very basic things you might wanna know:
About the scrolls:
The Elder Scrolls are some kind of Scrolls that existed since ever,and on them is written everything that happened,that is happening and that will happen.
They hold supernatural powers and for centuries where kept under heavy security in the Imperial City,and the only ones that had access to them where Moth priests.
But when the Imperial City was attacked and the White Gold Tower that holded the Scrolls was looted by the Thalmor,the Scrolls where spread.We don't know where they are now.
About the Thalmor:
The Thalmor are an organization of High Elves that made a militairy coup in Summerset Isle,and then started plotting against the Empire.
They sent agents to manipulate and indoctrinate the other races and managed to make Argonians and Khajit to leave the Empire.
Their plans where helped somewhat from a physical disaster too,when a commet fell on Vanderfell and the Volcano errupted killing many of the Dark Elf population.
The Thalmor want to take over the world,started war against the Empire,and signed a peace treaty with it,with rules that are considered by many Nords as treacherous.
The 9 (or 8) Divines and the Daedra:
The 9 divines and the Daedra are of the same form but there is a difference between them.
The 8 divines are also known as "Aedra" and Aedra where those that took part in the creation of the mortal world. Mortals thank the 8 divines because they created them.
The 9th divine is Talos,a man who isn't an Aedra,but he was a Nord hero that through his actions achieved immortality and thus mortals started worshipping him.
Daedra have similar powers and attributes with Aedra,but when the Aedra decided to build the mortal world,Daedra denied to take part in it.
Daedra are concieved as demons by the people of Tamriel.
The Dark Brotherhood:
Killers who are paid for murder contracts..
They like spooky places and gothic clothes.
There are more things,but you'd better go to uesp.net and start reading wiki pages there.
It's a good site.