Is Skyrim's story related to the previous Elder Scrolls games?

Skyrim takes place 200 years after the end of the last game, Oblivion.

Because it takes place so far from the rest of the games in the series (both temporally and geographically), there are few characters or locations that are relevant to any of the past games.

That doesn't mean that Skyrim has nothing to offer the stalwart Elder Scrolls fan, though.

The Dark Brotherhood, for instance, a player-joinable group of assassins-for-hire, returns in Skyrim, as do the Daedric Princes (or at least minor shrines devoted to them, manned by their followers.)

Even Alduin, the big-bad of Skyrim shows up in the lore of previous Elder Scrolls games, though the rest of the empire thinks he's the Aedra Akatosh. (He's actually his firstborn son)

If you're looking for a general history of Tamriel, the realm in which The Elder Scrolls takes place, the UESP has you covered there too.

But most of that is ancillary to the game itself. Skyrim was not designed as the 5th game in an ongoing series, but as a game able to stand on its own two feet. Even if you've never set foot on Nirn before, Skyrim will welcome you into its lore all the same!

There is a loose connection to most of the other Elder Scroll games. In Skyrim, you can obtain a book called "The Book of Dragonborn". The very last page of text has this prophecy:

When misrule takes its place at the eight corners of the world

When the Brass Tower walks and Time is reshaped

When the thrice-blessed fail and the Red Tower trembles

When the Dragonborn Ruler loses his throne, and the White Tower falls

When the Snow Tower lies sundered, kingless, bleeding

The World-Eater wakes, and the Wheel turns upon the Last Dragonborn.

Line 1 refers to the events in Arena (Jagar Tharn imprisoning the emperor and impersonating him ("misrule") and the eight pieces of the Staff of Chaos).

Line 2 refers to events in Daggerfall (Brass Tower refers to the Numidium golem).

Line 3 refers to events in Morrowind (the Red Tower is a reference to Red Mountain I think).

Line 4 refers to events in Oblivion (the Septim line fails and the White Tower in Cyrodiil is destroyed in the climactic battle).

Line 5 refers to current events in Skyrim (the murder of the High King of Skyrim).

So it appears the past events of the previous Elder Scrolls have all been leading up to the current situation in Skyrim, where you, the Last Dragonborn (as far as the Graybeards know of), must face off against the World-Eater.

There are a lot of tiny details that create a great feeling of connection. For example, after completing Dark Brotherhood storylime,

I'm riding the same horse and using the same blade that my character in Oblivion did.